Explanation:
Okay, so I need to figure out how many ways we can form a subcommittee of 5 members from a Senate committee that has 8 Republicans and 6 Democrats. The catch is that the subcommittee must have at least one member from each party. Hmm, let me think about how to approach this.

First, I remember that when dealing with combinations and restrictions like "at least one from each party," sometimes it's easier to calculate the total number of possible subcommittees without any restrictions and then subtract the ones that don't meet the requirement. That is, subtract the subcommittees that are all Republicans or all Democrats. That should leave me with the number of subcommittees that have at least one from each party. Let me check if that makes sense.

So, the total number of ways to choose 5 members from the entire committee, regardless of party, would be the combination of 14 people taken 5 at a time. Wait, the committee has 8 Republicans and 6 Democrats, so total members are 8 + 6 = 14. Yeah, so total ways is C(14, 5). Then, subtract the number of all-Republican subcommittees and all-Democrat subcommittees.

But let me make sure I'm not missing anything. The problem says "at least one member from each party," so that means we can't have all Republicans or all Democrats. So, yes, subtracting those two cases from the total should give the correct number.

Let me write that down:

Number of valid subcommittees = Total subcommittees - All-Republican subcommittees - All-Democrat subcommittees.

So, first, compute C(14, 5). Then compute C(8, 5) for all Republicans and C(6, 5) for all Democrats. Then subtract those two from the total.

But let me verify if that's the right approach. Another way to think about it is to consider the different possible distributions of Republicans and Democrats in the subcommittee. Since we need at least one from each party, the possible splits are:

- 1 Republican and 4 Democrats
- 2 Republicans and 3 Democrats
- 3 Republicans and 2 Democrats
- 4 Republicans and 1 Democrat

So, maybe I can calculate each of these cases separately and add them up. Let me try both methods and see if they give the same answer. That way, I can check if I did it right.

First, let me do the first method: total minus invalid.

Total number of subcommittees: C(14, 5). Let me compute that.

C(14, 5) is 14! / (5! * (14 - 5)!) = (14 * 13 * 12 * 11 * 10) / (5 * 4 * 3 * 2 * 1). Let me calculate that step by step.

First, numerator: 14*13=182, 182*12=2184, 2184*11=24024, 24024*10=240240.

Denominator: 5*4=20, 20*3=60, 60*2=120, 120*1=120.

So, 240240 / 120. Let me divide 240240 by 120. Dividing numerator and denominator by 10: 24024 / 12. Then divide numerator and denominator by 12: 24024 / 12 = 2002. Wait, 12*2000=24000, so 24024 - 24000 = 24, so 24 / 12 = 2. So total is 2000 + 2 = 2002. So C(14,5) is 2002. Okay.

Now, all-Republican subcommittees: C(8,5). Let me compute that. C(8,5) is the same as C(8,3) because of the combination identity C(n,k) = C(n, n - k). So C(8,3) = 8! / (3! * 5!) = (8*7*6)/(3*2*1) = 56. Let me check: 8*7=56, 56*6=336, divided by 6 (since 3! is 6) gives 56. Yes, that's right.

All-Democrat subcommittees: C(6,5). That's the same as C(6,1) = 6. So, 6.

Therefore, total invalid subcommittees are 56 + 6 = 62. Therefore, the number of valid subcommittees is 2002 - 62 = 1940.

Wait, so according to this method, the answer is 1940. Let me check with the other method, where I consider the different splits.

So, the possible splits are:

1R and 4D: C(8,1)*C(6,4)

2R and 3D: C(8,2)*C(6,3)

3R and 2D: C(8,3)*C(6,2)

4R and 1D: C(8,4)*C(6,1)

So, let me compute each of these and add them up.

First, 1R and 4D:

C(8,1) is 8. C(6,4) is C(6,2) = 15. So 8*15 = 120.

Second, 2R and 3D:

C(8,2) is 28. C(6,3) is 20. So 28*20 = 560.

Third, 3R and 2D:

C(8,3) is 56. C(6,2) is 15. So 56*15 = 840.

Fourth, 4R and 1D:

C(8,4) is 70. C(6,1) is 6. So 70*6 = 420.

Now, adding all these up: 120 + 560 + 840 + 420.

Let me compute step by step:

120 + 560 = 680

680 + 840 = 1520

1520 + 420 = 1940.

Same result! So that's reassuring. So both methods give 1940. Therefore, the answer is 1940.

But just to make sure I didn't make any calculation errors, let me check each of the combination numbers again.

First, C(8,1) is 8, correct. C(6,4) is 15. 8*15=120. Correct.

C(8,2) is 28. Let me verify: 8*7/2=28. Correct. C(6,3) is 20. 6*5*4/(3*2*1)= 120/6=20. Correct. 28*20=560. Correct.

C(8,3)=56. 8*7*6/(3*2*1)= 336/6=56. Correct. C(6,2)=15. 6*5/2=15. Correct. 56*15=840. Correct.

C(8,4)=70. Let me check: C(8,4)=70. Yes, because C(8,4)=70. And C(6,1)=6. 70*6=420. Correct. So adding up 120+560+840+420=1940. Correct.

Alternatively, the first method: total C(14,5)=2002. All Republicans C(8,5)=56. All Democrats C(6,5)=6. 56+6=62. 2002-62=1940. Correct. So both methods agree.

Therefore, the answer is 1940.

But just to be thorough, let me check if there's another way someone might approach this problem and maybe get a different answer, but I don't think so. For example, someone might think that "at least one from each party" means exactly one from each party, but that's not the case. The problem says "at least one," so it can be more. So the two methods I used are correct.

Alternatively, someone might miscalculate the combinations. For example, if they thought that C(14,5) is something else. Let me check that again.

C(14,5) is 2002. Let me check with another formula. For example, 14 choose 5:

14*13*12*11*10 / 5*4*3*2*1. Let me compute numerator and denominator again:

Numerator: 14*13=182, 182*12=2184, 2184*11=24024, 24024*10=240240.

Denominator: 5*4=20, 20*3=60, 60*2=120, 120*1=120.

So 240240 divided by 120. Let me divide 240240 by 10 first, which is 24024. Then divide by 12: 24024 / 12.

Dividing 24024 by 12: 12*2000=24000, so 24024 - 24000 = 24. 24 / 12 = 2. So total is 2000 + 2 = 2002. Correct.

So that's correct.

Another check: C(8,5)=56. Let me compute that again. 8 choose 5 is same as 8 choose 3. 8*7*6 / 3*2*1 = 336 / 6 = 56. Correct.

C(6,5)=6. Correct.

So 56 + 6 = 62. 2002 - 62 = 1940. Correct.

Therefore, I think the answer is definitely 1940.

But just to make sure, let me think if there's any other possible mistake. For example, maybe I misread the problem? Let me check again.

"A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee of 5 members that has at least one member from each party?"

Yes, so 8 R, 6 D. Subcommittees of 5, with at least one from each party. So yes, the answer is 1940.

Alternatively, if someone thought that "at least one from each party" means exactly one from each party, but that would be a different problem. But the question says "at least one," so that's not the case.

Alternatively, someone might have thought that the subcommittee has to have exactly one from each party, but that would be 1R and 1D, but since the subcommittee is of size 5, that's not possible. So that's not the case.

Alternatively, someone might have miscalculated the combinations for the different splits. For example, if they thought that C(6,4) is something else. Let me check C(6,4).

C(6,4) is 15. Let me compute that again. 6*5*4*3 / 4*3*2*1 = (360)/24 = 15. Correct.

Similarly, C(6,3)=20. Correct.

C(6,2)=15. Correct.

C(6,1)=6. Correct.

C(8,4)=70. Let me check that. 8*7*6*5 / 4*3*2*1 = 1680 / 24 = 70. Correct.

So all the numbers check out.

Therefore, I think I can be confident that the answer is 1940.

**Final Answer**
The number of ways is \boxed{1940}.

To determine the number of ways to form a subcommittee of 5 members from a Senate committee consisting of 8 Republicans and 6 Democrats, with the condition that the subcommittee must include **at least one member from each party**, we can approach the problem using **combinatorics**.

---

### Step 1: Total Number of Subcommittees (Without Restrictions)

We begin by calculating the total number of ways to choose 5 members from the entire committee of 14 people (8 Republicans + 6 Democrats):

$$
\binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002
$$

---

### Step 2: Subtract Invalid Subcommittees

We now subtract the number of subcommittees that **do not meet the requirement** of having at least one member from each party. These are the subcommittees that are **all Republicans** or **all Democrats**.

- **All Republicans**: Choose 5 Republicans from 8:
  $$
  \binom{8}{5} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
  $$

- **All Democrats**: Choose 5 Democrats from 6:
  $$
  \binom{6}{5} = \binom{6}{1} = 6
  $$

So, the total number of invalid subcommittees is:
$$
56 + 6 = 62
$$

---

### Step 3: Valid Subcommittees (At Least One from Each Party)

Now, subtract the invalid subcommittees from the total:
$$
2002 - 62 = 1940
$$

---

### Step 4: Verification via Case Analysis

As a cross-check, we can compute the number of valid subcommittees by considering all possible distributions of Republicans and Democrats that satisfy the "at least one from each party" condition. These are:

- 1R and 4D: $ \binom{8}{1} \times \binom{6}{4} = 8 \times 15 = 120 $
- 2R and 3D: $ \binom{8}{2} \times \binom{6}{3} = 28 \times 20 = 560 $
- 3R and 2D: $ \binom{8}{3} \times \binom{6}{2} = 56 \times 15 = 840 $
- 4R and 1D: $ \binom{8}{4} \times \binom{6}{1} = 70 \times 6 = 420 $

Adding these:
$$
120 + 560 + 840 + 420 = 1940
$$

This confirms the result.

---

### Final Answer

$$
\boxed{1940}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many ways we can form a subcommittee of 5 members from a Senate committee that has 8 Republicans and 6 Democrats. The catch is that the subcommittee must have at least one member from each party. Hmm, let me think about how to approach this.

First, I remember that when dealing with combinations and restrictions like "at least one from each party," sometimes it's easier to calculate the total number of possible subcommittees without any restrictions and then subtract the ones that don't meet the requirement. That is, subtract the subcommittees that are all Republicans or all Democrats. That should leave me with the number of subcommittees that have at least one from each party. Let me check if that makes sense.

So, the total number of ways to choose 5 members from the entire committee, regardless of party, would be the combination of 14 people taken 5 at a time. Wait, the committee has 8 Republicans and 6 Democrats, so total members are 8 + 6 = 14. Yeah, so total ways is C(14, 5). Then, subtract the number of all-Republican subcommittees and all-Democrat subcommittees.

But let me make sure I'm not missing anything. The problem says "at least one member from each party," so that means we can't have all Republicans or all Democrats. So, yes, subtracting those two cases from the total should give the correct number.

Let me write that down:

Number of valid subcommittees = Total subcommittees - All-Republican subcommittees - All-Democrat subcommittees.

So, first, compute C(14, 5). Then compute C(8, 5) for all Republicans and C(6, 5) for all Democrats. Then subtract those two from the total.

But let me verify if that's the right approach. Another way to think about it is to consider the different possible distributions of Republicans and Democrats in the subcommittee. Since we need at least one from each party, the possible splits are:

- 1 Republican and 4 Democrats
- 2 Republicans and 3 Democrats
- 3 Republicans and 2 Democrats
- 4 Republicans and 1 Democrat

So, maybe I can calculate each of these cases separately and add them up. Let me try both methods and see if they give the same answer. That way, I can check if I did it right.

First, let me do the first method: total minus invalid.

Total number of subcommittees: C(14, 5). Let me compute that.

C(14, 5) is 14! / (5! * (14 - 5)!) = (14 * 13 * 12 * 11 * 10) / (5 * 4 * 3 * 2 * 1). Let me calculate that step by step.

First, numerator: 14*13=182, 182*12=2184, 2184*11=24024, 24024*10=240240.

Denominator: 5*4=20, 20*3=60, 60*2=120, 120*1=120.

So, 240240 / 120. Let me divide 240240 by 120. Dividing numerator and denominator by 10: 24024 / 12. Then divide numerator and denominator by 12: 24024 / 12 = 2002. Wait, 12*2000=24000, so 24024 - 24000 = 24, so 24 / 12 = 2. So total is 2000 + 2 = 2002. So C(14,5) is 2002. Okay.

Now, all-Republican subcommittees: C(8,5). Let me compute that. C(8,5) is the same as C(8,3) because of the combination identity C(n,k) = C(n, n - k). So C(8,3) = 8! / (3! * 5!) = (8*7*6)/(3*2*1) = 56. Let me check: 8*7=56, 56*6=336, divided by 6 (since 3! is 6) gives 56. Yes, that's right.

All-Democrat subcommittees: C(6,5). That's the same as C(6,1) = 6. So, 6.

Therefore, total invalid subcommittees are 56 + 6 = 62. Therefore, the number of valid subcommittees is 2002 - 62 = 1940.

Wait, so according to this method, the answer is 1940. Let me check with the other method, where I consider the different splits.

So, the possible splits are:

1R and 4D: C(8,1)*C(6,4)

2R and 3D: C(8,2)*C(6,3)

3R and 2D: C(8,3)*C(6,2)

4R and 1D: C(8,4)*C(6,1)

So, let me compute each of these and add them up.

First, 1R and 4D:

C(8,1) is 8. C(6,4) is C(6,2) = 15. So 8*15 = 120.

Second, 2R and 3D:

C(8,2) is 28. C(6,3) is 20. So 28*20 = 560.

Third, 3R and 2D:

C(8,3) is 56. C(6,2) is 15. So 56*15 = 840.

Fourth, 4R and 1D:

C(8,4) is 70. C(6,1) is 6. So 70*6 = 420.

Now, adding all these up: 120 + 560 + 840 + 420.

Let me compute step by step:

120 + 560 = 680

680 + 840 = 1520

1520 + 420 = 1940.

Same result! So that's reassuring. So both methods give 1940. Therefore, the answer is 1940.

But just to make sure I didn't make any calculation errors, let me check each of the combination numbers again.

First, C(8,1) is 8, correct. C(6,4) is 15. 8*15=120. Correct.

C(8,2) is 28. Let me verify: 8*7/2=28. Correct. C(6,3) is 20. 6*5*4/(3*2*1)= 120/6=20. Correct. 28*20=560. Correct.

C(8,3)=56. 8*7*6/(3*2*1)= 336/6=56. Correct. C(6,2)=15. 6*5/2=15. Correct. 56*15=840. Correct.

C(8,4)=70. Let me check: C(8,4)=70. Yes, because C(8,4)=70. And C(6,1)=6. 70*6=420. Correct. So adding up 120+560+840+420=1940. Correct.

Alternatively, the first method: total C(14,5)=2002. All Republicans C(8,5)=56. All Democrats C(6,5)=6. 56+6=62. 2002-62=1940. Correct. So both methods agree.

Therefore, the answer is 1940.

But just to be thorough, let me check if there's another way someone might approach this problem and maybe get a different answer, but I don't think so. For example, someone might think that "at least one from each party" means exactly one from each party, but that's not the case. The problem says "at least one," so it can be more. So the two methods I used are correct.

Alternatively, someone might miscalculate the combinations. For example, if they thought that C(14,5) is something else. Let me check that again.

C(14,5) is 2002. Let me check with another formula. For example, 14 choose 5:

14*13*12*11*10 / 5*4*3*2*1. Let me compute numerator and denominator again:

Numerator: 14*13=182, 182*12=2184, 2184*11=24024, 24024*10=240240.

Denominator: 5*4=20, 20*3=60, 60*2=120, 120*1=120.

So 240240 divided by 120. Let me divide 240240 by 10 first, which is 24024. Then divide by 12: 24024 / 12.

Dividing 24024 by 12: 12*2000=24000, so 24024 - 24000 = 24. 24 / 12 = 2. So total is 2000 + 2 = 2002. Correct.

So that's correct.

Another check: C(8,5)=56. Let me compute that again. 8 choose 5 is same as 8 choose 3. 8*7*6 / 3*2*1 = 336 / 6 = 56. Correct.

C(6,5)=6. Correct.

So 56 + 6 = 62. 2002 - 62 = 1940. Correct.

Therefore, I think the answer is definitely 1940.

But just to make sure, let me think if there's any other possible mistake. For example, maybe I misread the problem? Let me check again.

"A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee of 5 members that has at least one member from each party?"

Yes, so 8 R, 6 D. Subcommittees of 5, with at least one from each party. So yes, the answer is 1940.

Alternatively, if someone thought that "at least one from each party" means exactly one from each party, but that would be a different problem. But the question says "at least one," so that's not the case.

Alternatively, someone might have thought that the subcommittee has to have exactly one from each party, but that would be 1R and 1D, but since the subcommittee is of size 5, that's not possible. So that's not the case.

Alternatively, someone might have miscalculated the combinations for the different splits. For example, if they thought that C(6,4) is something else. Let me check C(6,4).

C(6,4) is 15. Let me compute that again. 6*5*4*3 / 4*3*2*1 = (360)/24 = 15. Correct.

Similarly, C(6,3)=20. Correct.

C(6,2)=15. Correct.

C(6,1)=6. Correct.

C(8,4)=70. Let me check that. 8*7*6*5 / 4*3*2*1 = 1680 / 24 = 70. Correct.

So all the numbers check out.

Therefore, I think I can be confident that the answer is 1940.

**Final Answer**
The number of ways is \boxed{1940}.

To determine the number of ways to form a subcommittee of 5 members from a Senate committee consisting of 8 Republicans and 6 Democrats, with the condition that the subcommittee must include **at least one member from each party**, we can approach the problem using **combinatorics**.

---

### Step 1: Total Number of Subcommittees (Without Restrictions)

We begin by calculating the total number of ways to choose 5 members from the entire committee of 14 people (8 Republicans + 6 Democrats):

$$
\binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002
$$

---

### Step 2: Subtract Invalid Subcommittees

We now subtract the number of subcommittees that **do not meet the requirement** of having at least one member from each party. These are the subcommittees that are **all Republicans** or **all Democrats**.

- **All Republicans**: Choose 5 Republicans from 8:
  $$
  \binom{8}{5} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
  $$

- **All Democrats**: Choose 5 Democrats from 6:
  $$
  \binom{6}{5} = \binom{6}{1} = 6
  $$

So, the total number of invalid subcommittees is:
$$
56 + 6 = 62
$$

---

### Step 3: Valid Subcommittees (At Least One from Each Party)

Now, subtract the invalid subcommittees from the total:
$$
2002 - 62 = 1940
$$

---

### Step 4: Verification via Case Analysis

As a cross-check, we can compute the number of valid subcommittees by considering all possible distributions of Republicans and Democrats that satisfy the "at least one from each party" condition. These are:

- 1R and 4D: $ \binom{8}{1} \times \binom{6}{4} = 8 \times 15 = 120 $
- 2R and 3D: $ \binom{8}{2} \times \binom{6}{3} = 28 \times 20 = 560 $
- 3R and 2D: $ \binom{8}{3} \times \binom{6}{2} = 56 \times 15 = 840 $
- 4R and 1D: $ \binom{8}{4} \times \binom{6}{1} = 70 \times 6 = 420 $

Adding these:
$$
120 + 560 + 840 + 420 = 1940
$$

This confirms the result.

---

### Final Answer

$$
\boxed{1940}
$$

Solution: There are a total of $\binom{14}{5}=2002$ ways of selecting a subcommittee of 5 with no restrictions on the membership. Of these committees, the only ones that will violate the given condition are the ones that consist entirely of Republicans or entirely of Democrats. There are $\binom{8}{5}=56$ possible subcommittees that have all 5 members selected from among the 8 Republicans and $\binom{6}{5}=6$ possible subcommittees that have all 5 members selected from among the 6 Democrats. Subtracting the number of subcommittees that don't work from the total number of possible subcommittees gives us our answer: $2002-56-6=\boxed{1940}$.


Explanation:
Okay, so I need to simplify this expression: the square root of a fraction, where the numerator is the sum of three terms: 5 over the square root of 80, plus the square root of 845 over 9, plus the square root of 45. And all of that is divided by the square root of 5. Then take the square root of the whole thing. Hmm, that sounds complicated, but maybe if I break it down step by step, it will make more sense.

First, let me write down the expression again to make sure I have it right:

√[( (5/√80) + (√845)/9 + √45 ) / √5]

Alright, so the main structure is square root of [ (something) divided by √5 ]. Maybe I can simplify the numerator first before dealing with the denominator. Let me focus on simplifying each term in the numerator individually.

Starting with the first term: 5 divided by √80. I know that square roots can often be simplified by factoring out perfect squares. Let me see if I can simplify √80. 80 factors into 16 * 5, right? Because 16 is a perfect square. So √80 = √(16*5) = √16 * √5 = 4√5. So then 5 divided by √80 is 5 divided by 4√5. Let me write that as 5/(4√5). Maybe I can simplify this further by rationalizing the denominator? Let me try that.

Multiply numerator and denominator by √5: [5 * √5] / [4√5 * √5] = [5√5] / [4*5] = [5√5]/20 = √5/4. Oh, that's simpler! So the first term simplifies to √5/4. Okay, that's good progress.

Now moving on to the second term: √845 divided by 9. Let me simplify √845. Let me factor 845. Hmm, 845 divided by 5 is 169. Oh, 169 is 13 squared. So √845 = √(5*13²) = √(13² * 5) = 13√5. Therefore, √845 / 9 is 13√5 / 9. That seems straightforward. So the second term is 13√5 over 9.

Third term is √45. Let me simplify that. 45 is 9*5, so √45 = √(9*5) = √9 * √5 = 3√5. So the third term is 3√5.

So now, putting all the simplified terms back into the numerator:

First term: √5/4

Second term: 13√5/9

Third term: 3√5

So the numerator is √5/4 + 13√5/9 + 3√5. Now, since all terms have √5, maybe I can factor that out? Let me try that.

Let me write each term with √5:

√5*(1/4 + 13/9 + 3). Wait, but the third term is 3√5, which is 3√5/1. So to combine them, I need a common denominator for the fractions. Let me compute the sum inside the parentheses.

First, let me compute 1/4 + 13/9 + 3. Let me convert all terms to fractions with denominator... Let me see. The denominators are 4, 9, and 1. The least common denominator of 4 and 9 is 36. So let me convert each term:

1/4 = 9/36

13/9 = (13*4)/36 = 52/36

3 is 3/1 = (3*36)/36 = 108/36

So adding them up: 9/36 + 52/36 + 108/36 = (9 + 52 + 108)/36 = let me compute that.

9 + 52 is 61, plus 108 is 169. So total is 169/36. Therefore, the numerator simplifies to √5*(169/36). So the entire numerator is (169√5)/36.

So now, going back to the original expression, the entire numerator is (169√5)/36, and then we have to divide that by √5. So the whole expression inside the square root is [(169√5)/36] divided by √5. Let me write that as:

[(169√5)/36] / √5 = (169√5)/36 * 1/√5 = 169√5 / (36√5)

Now, √5 divided by √5 is 1, so this simplifies to 169/36. Wait, really? Let me check that again.

Yes, because √5 divided by √5 is 1, so the √5 cancels out. Therefore, the entire expression inside the square root simplifies to 169/36. Therefore, the original expression is the square root of 169/36.

Now, square root of 169 is 13, and square root of 36 is 6. Therefore, √(169/36) = 13/6. So the simplified fraction is 13/6.

Wait, let me just recap all steps to make sure I didn't make a mistake. Starting with the first term: 5 over sqrt(80). Simplified sqrt(80) to 4 sqrt(5), so 5/(4 sqrt(5)) becomes sqrt(5)/4. Then second term sqrt(845)/9. sqrt(845) is sqrt(169*5) which is 13 sqrt(5), so 13 sqrt(5)/9. Third term sqrt(45) is 3 sqrt(5). Then adding them up: sqrt(5)/4 + 13 sqrt(5)/9 + 3 sqrt(5). Factored out sqrt(5) and converted to common denominator 36. Got 1/4 as 9/36, 13/9 as 52/36, and 3 as 108/36. Added to 169/36. Then multiplied by sqrt(5) gives 169 sqrt(5)/36. Divided by sqrt(5) gives 169/36. Square root of that is 13/6. Seems correct.

But let me check if there's any step where I might have miscalculated. Let me verify the addition of 9 + 52 + 108. 9 + 52 is 61, 61 + 108 is 169. Yes, that's correct. 169 is 13 squared, so that's good. Then 169/36 under the square root. Square root of 169 is 13, square root of 36 is 6. So 13/6. So the simplified fraction is 13/6. That seems right.

But let me check if I handled the division by sqrt(5) correctly. The numerator was (169√5)/36 divided by √5. So dividing by √5 is multiplying by 1/√5. So (169√5)/36 * 1/√5 = 169/36 * (√5 / √5) = 169/36 * 1 = 169/36. Yes, that's correct. So that step is okay.

Alternatively, maybe I can think of the entire expression as:

Original expression: sqrt( [ (5/sqrt(80) + sqrt(845)/9 + sqrt(45)) / sqrt(5) ] )

Alternatively, maybe I can combine the terms differently? Let me try another approach to verify.

Alternatively, maybe I can rationalize each term first and then combine. Wait, but I think my previous steps are correct. Let me check each term again.

First term: 5 / sqrt(80). sqrt(80) is 4 sqrt(5). So 5 / (4 sqrt(5)) = (5 / 4) * (1 / sqrt(5)) = (5 / 4) * sqrt(5)/5 = sqrt(5)/4. Yes, that's correct. So that term is correct.

Second term: sqrt(845)/9. sqrt(845) is sqrt(169*5) = 13 sqrt(5). So 13 sqrt(5)/9. Correct.

Third term: sqrt(45) is 3 sqrt(5). Correct.

Adding them up: sqrt(5)/4 + 13 sqrt(5)/9 + 3 sqrt(5). Let me convert them all to decimals to check? Maybe that's overkill, but just to verify.

Let me compute each term numerically:

First term: sqrt(5)/4 ≈ 2.236 / 4 ≈ 0.559

Second term: 13 sqrt(5)/9 ≈ 13 * 2.236 / 9 ≈ 29.068 / 9 ≈ 3.2298

Third term: 3 sqrt(5) ≈ 3 * 2.236 ≈ 6.708

Adding them up: 0.559 + 3.2298 + 6.708 ≈ 0.559 + 3.2298 is 3.7888 + 6.708 ≈ 10.4968

Now, let me compute the entire numerator divided by sqrt(5):

So numerator ≈ 10.4968, divided by sqrt(5) ≈ 2.236. So 10.4968 / 2.236 ≈ let me compute that. 2.236 * 4.7 ≈ 2.236 * 4 + 2.236 * 0.7 ≈ 8.944 + 1.565 ≈ 10.509. Which is very close to 10.4968. So approximately 4.7. Then taking the square root of that would be sqrt(4.7) ≈ 2.167. Now, 13/6 is approximately 2.166..., which matches. So that numerical check seems to confirm that the answer is 13/6. So that gives me more confidence.

Alternatively, if I compute 169/36, that is approximately 4.694, whose square root is approx 2.166, which is 13/6. So that's correct.

Therefore, I think my answer is correct. So the simplified fraction is 13/6.

**Final Answer**
\boxed{\dfrac{13}{6}}

To simplify the expression:

$$
\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt{5}}}
$$

we proceed step by step, simplifying each term and combining them appropriately.

---

### Step 1: Simplify Each Term in the Numerator

#### First Term: $\dfrac{5}{\sqrt{80}}$

We simplify $\sqrt{80}$:

$$
\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}
$$

So:

$$
\dfrac{5}{\sqrt{80}} = \dfrac{5}{4\sqrt{5}} = \dfrac{5}{4\sqrt{5}} \cdot \dfrac{\sqrt{5}}{\sqrt{5}} = \dfrac{5\sqrt{5}}{20} = \dfrac{\sqrt{5}}{4}
$$

#### Second Term: $\dfrac{\sqrt{845}}{9}$

We simplify $\sqrt{845}$:

$$
\sqrt{845} = \sqrt{169 \cdot 5} = 13\sqrt{5}
$$

So:

$$
\dfrac{\sqrt{845}}{9} = \dfrac{13\sqrt{5}}{9}
$$

#### Third Term: $\sqrt{45}$

$$
\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}
$$

---

### Step 2: Combine the Simplified Terms

Now, the numerator becomes:

$$
\dfrac{\sqrt{5}}{4} + \dfrac{13\sqrt{5}}{9} + 3\sqrt{5}
$$

Factor out $\sqrt{5}$:

$$
\sqrt{5} \left( \dfrac{1}{4} + \dfrac{13}{9} + 3 \right)
$$

We now compute the sum inside the parentheses. To do this, find a common denominator. The least common denominator of 4, 9, and 1 is 36.

Convert each term:

- $\dfrac{1}{4} = \dfrac{9}{36}$
- $\dfrac{13}{9} = \dfrac{52}{36}$
- $3 = \dfrac{108}{36}$

Add them:

$$
\dfrac{9}{36} + \dfrac{52}{36} + \dfrac{108}{36} = \dfrac{169}{36}
$$

So the numerator becomes:

$$
\sqrt{5} \cdot \dfrac{169}{36} = \dfrac{169\sqrt{5}}{36}
$$

---

### Step 3: Divide by $\sqrt{5}$

Now, the entire expression inside the square root is:

$$
\dfrac{\dfrac{169\sqrt{5}}{36}}{\sqrt{5}} = \dfrac{169\sqrt{5}}{36\sqrt{5}} = \dfrac{169}{36}
$$

(Since $\sqrt{5}/\sqrt{5} = 1$)

---

### Step 4: Take the Square Root

$$
\sqrt{\dfrac{169}{36}} = \dfrac{\sqrt{169}}{\sqrt{36}} = \dfrac{13}{6}
$$

---

### Final Answer

$$
\boxed{\dfrac{13}{6}}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: the square root of a fraction, where the numerator is the sum of three terms: 5 over the square root of 80, plus the square root of 845 over 9, plus the square root of 45. And all of that is divided by the square root of 5. Then take the square root of the whole thing. Hmm, that sounds complicated, but maybe if I break it down step by step, it will make more sense.

First, let me write down the expression again to make sure I have it right:

√[( (5/√80) + (√845)/9 + √45 ) / √5]

Alright, so the main structure is square root of [ (something) divided by √5 ]. Maybe I can simplify the numerator first before dealing with the denominator. Let me focus on simplifying each term in the numerator individually.

Starting with the first term: 5 divided by √80. I know that square roots can often be simplified by factoring out perfect squares. Let me see if I can simplify √80. 80 factors into 16 * 5, right? Because 16 is a perfect square. So √80 = √(16*5) = √16 * √5 = 4√5. So then 5 divided by √80 is 5 divided by 4√5. Let me write that as 5/(4√5). Maybe I can simplify this further by rationalizing the denominator? Let me try that.

Multiply numerator and denominator by √5: [5 * √5] / [4√5 * √5] = [5√5] / [4*5] = [5√5]/20 = √5/4. Oh, that's simpler! So the first term simplifies to √5/4. Okay, that's good progress.

Now moving on to the second term: √845 divided by 9. Let me simplify √845. Let me factor 845. Hmm, 845 divided by 5 is 169. Oh, 169 is 13 squared. So √845 = √(5*13²) = √(13² * 5) = 13√5. Therefore, √845 / 9 is 13√5 / 9. That seems straightforward. So the second term is 13√5 over 9.

Third term is √45. Let me simplify that. 45 is 9*5, so √45 = √(9*5) = √9 * √5 = 3√5. So the third term is 3√5.

So now, putting all the simplified terms back into the numerator:

First term: √5/4

Second term: 13√5/9

Third term: 3√5

So the numerator is √5/4 + 13√5/9 + 3√5. Now, since all terms have √5, maybe I can factor that out? Let me try that.

Let me write each term with √5:

√5*(1/4 + 13/9 + 3). Wait, but the third term is 3√5, which is 3√5/1. So to combine them, I need a common denominator for the fractions. Let me compute the sum inside the parentheses.

First, let me compute 1/4 + 13/9 + 3. Let me convert all terms to fractions with denominator... Let me see. The denominators are 4, 9, and 1. The least common denominator of 4 and 9 is 36. So let me convert each term:

1/4 = 9/36

13/9 = (13*4)/36 = 52/36

3 is 3/1 = (3*36)/36 = 108/36

So adding them up: 9/36 + 52/36 + 108/36 = (9 + 52 + 108)/36 = let me compute that.

9 + 52 is 61, plus 108 is 169. So total is 169/36. Therefore, the numerator simplifies to √5*(169/36). So the entire numerator is (169√5)/36.

So now, going back to the original expression, the entire numerator is (169√5)/36, and then we have to divide that by √5. So the whole expression inside the square root is [(169√5)/36] divided by √5. Let me write that as:

[(169√5)/36] / √5 = (169√5)/36 * 1/√5 = 169√5 / (36√5)

Now, √5 divided by √5 is 1, so this simplifies to 169/36. Wait, really? Let me check that again.

Yes, because √5 divided by √5 is 1, so the √5 cancels out. Therefore, the entire expression inside the square root simplifies to 169/36. Therefore, the original expression is the square root of 169/36.

Now, square root of 169 is 13, and square root of 36 is 6. Therefore, √(169/36) = 13/6. So the simplified fraction is 13/6.

Wait, let me just recap all steps to make sure I didn't make a mistake. Starting with the first term: 5 over sqrt(80). Simplified sqrt(80) to 4 sqrt(5), so 5/(4 sqrt(5)) becomes sqrt(5)/4. Then second term sqrt(845)/9. sqrt(845) is sqrt(169*5) which is 13 sqrt(5), so 13 sqrt(5)/9. Third term sqrt(45) is 3 sqrt(5). Then adding them up: sqrt(5)/4 + 13 sqrt(5)/9 + 3 sqrt(5). Factored out sqrt(5) and converted to common denominator 36. Got 1/4 as 9/36, 13/9 as 52/36, and 3 as 108/36. Added to 169/36. Then multiplied by sqrt(5) gives 169 sqrt(5)/36. Divided by sqrt(5) gives 169/36. Square root of that is 13/6. Seems correct.

But let me check if there's any step where I might have miscalculated. Let me verify the addition of 9 + 52 + 108. 9 + 52 is 61, 61 + 108 is 169. Yes, that's correct. 169 is 13 squared, so that's good. Then 169/36 under the square root. Square root of 169 is 13, square root of 36 is 6. So 13/6. So the simplified fraction is 13/6. That seems right.

But let me check if I handled the division by sqrt(5) correctly. The numerator was (169√5)/36 divided by √5. So dividing by √5 is multiplying by 1/√5. So (169√5)/36 * 1/√5 = 169/36 * (√5 / √5) = 169/36 * 1 = 169/36. Yes, that's correct. So that step is okay.

Alternatively, maybe I can think of the entire expression as:

Original expression: sqrt( [ (5/sqrt(80) + sqrt(845)/9 + sqrt(45)) / sqrt(5) ] )

Alternatively, maybe I can combine the terms differently? Let me try another approach to verify.

Alternatively, maybe I can rationalize each term first and then combine. Wait, but I think my previous steps are correct. Let me check each term again.

First term: 5 / sqrt(80). sqrt(80) is 4 sqrt(5). So 5 / (4 sqrt(5)) = (5 / 4) * (1 / sqrt(5)) = (5 / 4) * sqrt(5)/5 = sqrt(5)/4. Yes, that's correct. So that term is correct.

Second term: sqrt(845)/9. sqrt(845) is sqrt(169*5) = 13 sqrt(5). So 13 sqrt(5)/9. Correct.

Third term: sqrt(45) is 3 sqrt(5). Correct.

Adding them up: sqrt(5)/4 + 13 sqrt(5)/9 + 3 sqrt(5). Let me convert them all to decimals to check? Maybe that's overkill, but just to verify.

Let me compute each term numerically:

First term: sqrt(5)/4 ≈ 2.236 / 4 ≈ 0.559

Second term: 13 sqrt(5)/9 ≈ 13 * 2.236 / 9 ≈ 29.068 / 9 ≈ 3.2298

Third term: 3 sqrt(5) ≈ 3 * 2.236 ≈ 6.708

Adding them up: 0.559 + 3.2298 + 6.708 ≈ 0.559 + 3.2298 is 3.7888 + 6.708 ≈ 10.4968

Now, let me compute the entire numerator divided by sqrt(5):

So numerator ≈ 10.4968, divided by sqrt(5) ≈ 2.236. So 10.4968 / 2.236 ≈ let me compute that. 2.236 * 4.7 ≈ 2.236 * 4 + 2.236 * 0.7 ≈ 8.944 + 1.565 ≈ 10.509. Which is very close to 10.4968. So approximately 4.7. Then taking the square root of that would be sqrt(4.7) ≈ 2.167. Now, 13/6 is approximately 2.166..., which matches. So that numerical check seems to confirm that the answer is 13/6. So that gives me more confidence.

Alternatively, if I compute 169/36, that is approximately 4.694, whose square root is approx 2.166, which is 13/6. So that's correct.

Therefore, I think my answer is correct. So the simplified fraction is 13/6.

**Final Answer**
\boxed{\dfrac{13}{6}}

To simplify the expression:

$$
\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt{5}}}
$$

we proceed step by step, simplifying each term and combining them appropriately.

---

### Step 1: Simplify Each Term in the Numerator

#### First Term: $\dfrac{5}{\sqrt{80}}$

We simplify $\sqrt{80}$:

$$
\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}
$$

So:

$$
\dfrac{5}{\sqrt{80}} = \dfrac{5}{4\sqrt{5}} = \dfrac{5}{4\sqrt{5}} \cdot \dfrac{\sqrt{5}}{\sqrt{5}} = \dfrac{5\sqrt{5}}{20} = \dfrac{\sqrt{5}}{4}
$$

#### Second Term: $\dfrac{\sqrt{845}}{9}$

We simplify $\sqrt{845}$:

$$
\sqrt{845} = \sqrt{169 \cdot 5} = 13\sqrt{5}
$$

So:

$$
\dfrac{\sqrt{845}}{9} = \dfrac{13\sqrt{5}}{9}
$$

#### Third Term: $\sqrt{45}$

$$
\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}
$$

---

### Step 2: Combine the Simplified Terms

Now, the numerator becomes:

$$
\dfrac{\sqrt{5}}{4} + \dfrac{13\sqrt{5}}{9} + 3\sqrt{5}
$$

Factor out $\sqrt{5}$:

$$
\sqrt{5} \left( \dfrac{1}{4} + \dfrac{13}{9} + 3 \right)
$$

We now compute the sum inside the parentheses. To do this, find a common denominator. The least common denominator of 4, 9, and 1 is 36.

Convert each term:

- $\dfrac{1}{4} = \dfrac{9}{36}$
- $\dfrac{13}{9} = \dfrac{52}{36}$
- $3 = \dfrac{108}{36}$

Add them:

$$
\dfrac{9}{36} + \dfrac{52}{36} + \dfrac{108}{36} = \dfrac{169}{36}
$$

So the numerator becomes:

$$
\sqrt{5} \cdot \dfrac{169}{36} = \dfrac{169\sqrt{5}}{36}
$$

---

### Step 3: Divide by $\sqrt{5}$

Now, the entire expression inside the square root is:

$$
\dfrac{\dfrac{169\sqrt{5}}{36}}{\sqrt{5}} = \dfrac{169\sqrt{5}}{36\sqrt{5}} = \dfrac{169}{36}
$$

(Since $\sqrt{5}/\sqrt{5} = 1$)

---

### Step 4: Take the Square Root

$$
\sqrt{\dfrac{169}{36}} = \dfrac{\sqrt{169}}{\sqrt{36}} = \dfrac{13}{6}
$$

---

### Final Answer

$$
\boxed{\dfrac{13}{6}}
$$

Solution:
First, we'll divide $\sqrt{5}$ into each term in the numerator of the fraction inside the big radical: $$\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt5}}=
\sqrt{\frac{5}{\sqrt{80}\cdot\sqrt{5}} + \frac{\sqrt{845}}{9\sqrt{5}} + \frac{\sqrt{45}}{\sqrt{5}}}.
$$Let's attack each fraction within the square root separately. First, $$\dfrac{5}{\sqrt{80}\cdot\sqrt5}=\dfrac{5}{\sqrt{400}}=\dfrac{5}{20}=\dfrac{1}{4}.$$The second one is trickier: $$\dfrac{\sqrt{845}}{9\sqrt5}=\dfrac{\sqrt{169}}{9}=\dfrac{13}{9}.$$Finally, $\dfrac{\sqrt{45}}{\sqrt5}=\sqrt9=3$. Adding these together, we get $$\sqrt{\dfrac{1}{4}+\dfrac{13}{9}+3}=\sqrt{\dfrac{9+52+108}{36}}=\sqrt{\dfrac{169}{36}}=\boxed{\frac{13}{6}}.$$


Explanation:
Okay, so I need to find the minimum value of this expression:

$$
\frac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc}
$$

where $ a $, $ b $, and $ c $ are positive real numbers. Hmm, let me think about how to approach this. It looks like an optimization problem with three variables. Maybe I can use calculus, like taking partial derivatives and setting them to zero? But that might get complicated with three variables. Alternatively, maybe there's a way to apply inequalities like AM-GM or Cauchy-Schwarz? Let me try to simplify the expression first.

First, let me expand the numerator. Let me denote $ S = a + b + c $, but maybe that's not helpful yet. Let me look at the terms inside the brackets:

$$
(a + b)^2 + (a + b + 4c)^2
$$

Let me compute each term:

First term: $ (a + b)^2 = a^2 + 2ab + b^2 $

Second term: $ (a + b + 4c)^2 = (a + b)^2 + 8c(a + b) + 16c^2 $

So adding them together:

$$
(a + b)^2 + (a + b + 4c)^2 = [a^2 + 2ab + b^2] + [a^2 + 2ab + b^2 + 8c(a + b) + 16c^2]
$$

Combine like terms:

- $ a^2 + a^2 = 2a^2 $
- $ 2ab + 2ab = 4ab $
- $ b^2 + b^2 = 2b^2 $
- Then the other terms: $ 8c(a + b) + 16c^2 $

So altogether:

$$
2a^2 + 4ab + 2b^2 + 8c(a + b) + 16c^2
$$

Hmm, maybe factor some terms? Let me see:

First, notice that $ 2a^2 + 4ab + 2b^2 = 2(a^2 + 2ab + b^2) = 2(a + b)^2 $. So that simplifies part of it. Then the rest is $ 8c(a + b) + 16c^2 $. So the entire numerator becomes:

$$
(a + b + c) \times [2(a + b)^2 + 8c(a + b) + 16c^2]
$$

Let me write that as:

$$
(a + b + c) \times [2(a + b)^2 + 8c(a + b) + 16c^2]
$$

Hmm, maybe factor out a 2 from the bracketed terms? Let me check:

Wait, 2(a + b)^2 + 8c(a + b) + 16c^2. Let me factor 2 from the first two terms:

2[(a + b)^2 + 4c(a + b)] + 16c^2. Hmm, not sure if that helps. Alternatively, maybe set variables to simplify. Let me think.

Let me denote $ x = a + b $, and $ y = c $. Then the expression becomes:

Numerator: $ (x + y)[2x^2 + 8y x + 16y^2] $

Denominator: $ abc $. But $ a $ and $ b $ are still variables. However, since $ x = a + b $, maybe I can express $ ab $ in terms of $ x $ and another variable? For example, if I fix $ x = a + b $, then $ ab $ is maximized when $ a = b $, but since we are dealing with a ratio, maybe we need to find the minimum of the entire expression. Wait, but the denominator is $ abc $, so if I fix $ x = a + b $, then $ ab $ is related to $ a $ and $ b $, but $ c $ is separate. Maybe I need to consider variables substitution.

Alternatively, maybe use symmetry or set variables in terms of each other. Let me think. Let me try to assume some relation between a, b, c. For example, maybe set a = b? Since the expression is symmetric in a and b? Let me check.

Looking at the original expression:

The numerator has (a + b + c) multiplied by [(a + b)^2 + (a + b + 4c)^2]. The denominator is abc. Since the expression is symmetric in a and b, maybe the minimum occurs when a = b? Let me assume that a = b. Let me set a = b = t, and then express everything in terms of t and c. Let me try that.

Let me set a = b = t. Then:

Numerator:

First, a + b + c = 2t + c

Then, (a + b)^2 + (a + b + 4c)^2 = (2t)^2 + (2t + 4c)^2 = 4t² + (2t + 4c)²

Compute that:

(2t + 4c)² = 4t² + 16tc + 16c²

So total numerator inside the brackets: 4t² + 4t² + 16tc + 16c² = 8t² + 16tc + 16c²

Therefore, the entire numerator is (2t + c)(8t² + 16tc + 16c²)

Denominator is abc = t * t * c = t²c

So the entire expression becomes:

[ (2t + c)(8t² + 16tc + 16c²) ] / (t²c)

Let me simplify this expression. Let me factor numerator:

First, note that 8t² + 16tc + 16c² = 8(t² + 2tc + 2c²). Wait, let me check:

Wait 8t² + 16tc + 16c² = 8t² + 16tc + 16c². Let me factor 8 from the first term and 16 from the others? Not sure. Alternatively, factor 8:

Wait 8t² + 16tc + 16c² = 8(t² + 2tc) + 16c²? Not helpful. Alternatively, maybe factor 8t² + 16tc + 16c² = 8(t² + 2tc + 2c²). Wait, 8(t² + 2tc + 2c²) would be 8t² + 16tc + 16c². Yes, exactly. So that's correct.

So numerator becomes (2t + c) * 8(t² + 2tc + 2c²) = 8(2t + c)(t² + 2tc + 2c²)

Denominator is t²c. Therefore, the entire expression is:

8(2t + c)(t² + 2tc + 2c²) / (t²c)

Hmm, maybe simplify this expression. Let me denote variables in terms of ratios. Let me set k = t/c. Since t and c are positive, k is positive. Let me set k = t/c, so t = kc. Then substitute into the expression.

So t = kc. Then:

First, 2t + c = 2kc + c = c(2k + 1)

Next, t² + 2tc + 2c² = (k²c²) + 2(kc)c + 2c² = c²(k² + 2k + 2)

Therefore, numerator becomes 8 * c(2k + 1) * c²(k² + 2k + 2) = 8c³(2k + 1)(k² + 2k + 2)

Denominator is t²c = (k²c²)c = k²c³

Therefore, the entire expression becomes:

[8c³(2k + 1)(k² + 2k + 2)] / [k²c³] = 8(2k + 1)(k² + 2k + 2)/k²

So now, the problem reduces to minimizing the function f(k) = 8(2k + 1)(k² + 2k + 2)/k² where k > 0.

Okay, so now it's a single-variable calculus problem. Let me compute f(k):

First, expand the numerator:

(2k + 1)(k² + 2k + 2). Let me compute that:

Multiply term by term:

First, 2k*(k²) = 2k³

2k*(2k) = 4k²

2k*(2) = 4k

Then 1*(k²) = k²

1*(2k) = 2k

1*(2) = 2

So adding all together:

2k³ + 4k² + 4k + k² + 2k + 2 = 2k³ + (4k² + k²) + (4k + 2k) + 2 = 2k³ + 5k² + 6k + 2

Therefore, f(k) = 8*(2k³ + 5k² + 6k + 2)/k² = 8*(2k³/k² + 5k²/k² + 6k/k² + 2/k²) = 8*(2k + 5 + 6/k + 2/k²)

So f(k) = 8*(2k + 5 + 6/k + 2/k²)

Alternatively, maybe keep it as 8*(2k³ + 5k² + 6k + 2)/k². Either way, to find the minimum, take derivative with respect to k and set to zero.

Let me compute f(k) = 8*(2k³ + 5k² + 6k + 2)/k². Let me denote f(k) = 8*(2k³ + 5k² + 6k + 2)/k². Let me compute derivative f’(k):

First, let me write f(k) as 8*(2k³ + 5k² + 6k + 2)*k^{-2}

So derivative f’(k) = 8*[ derivative of (2k³ + 5k² + 6k + 2)*k^{-2} + (2k³ + 5k² + 6k + 2)* derivative of k^{-2} ]

Wait, actually, using product rule:

Let me denote u = 2k³ + 5k² + 6k + 2 and v = k^{-2}

Then f(k) = 8uv, so f’(k) = 8(u’v + uv’)

Compute u’:

u’ = 6k² + 10k + 6

v’ = -2k^{-3}

Therefore:

f’(k) = 8[ (6k² + 10k + 6)(k^{-2}) + (2k³ + 5k² + 6k + 2)(-2k^{-3}) ]

Let me compute each term:

First term: (6k² + 10k + 6)/k² = 6 + 10/k + 6/k²

Second term: -2(2k³ + 5k² + 6k + 2)/k³ = -2*(2 + 5/k + 6/k² + 2/k³)

Therefore, combining:

f’(k) = 8[ (6 + 10/k + 6/k²) - 2*(2 + 5/k + 6/k² + 2/k³) ]

Let me compute inside the brackets:

First expand the second part:

-2*(2 + 5/k + 6/k² + 2/k³) = -4 -10/k -12/k² -4/k³

Now add the first part:

(6 + 10/k + 6/k²) + (-4 -10/k -12/k² -4/k³) =

Compute term by term:

6 - 4 = 2

10/k -10/k = 0

6/k² -12/k² = -6/k²

Then -4/k³

So overall: 2 - 6/k² - 4/k³

Therefore, f’(k) = 8*(2 - 6/k² - 4/k³) = 16 - 48/k² - 32/k³

Set derivative equal to zero:

16 - 48/k² - 32/k³ = 0

Multiply both sides by k³ to eliminate denominators:

16k³ - 48k - 32 = 0

Simplify:

Divide both sides by 16:

k³ - 3k - 2 = 0

So we have the equation k³ - 3k - 2 = 0. Let me solve this cubic equation.

Let me try rational roots. Possible rational roots are ±1, ±2.

Test k=1: 1 - 3 - 2 = -4 ≠ 0

k=-1: -1 + 3 - 2 = 0. Oh, k=-1 is a root. Therefore, we can factor (k + 1) from the cubic.

Using polynomial division or synthetic division:

Divide k³ - 3k - 2 by (k + 1). Let me do synthetic division:

Coefficients: 1 (k³), 0 (k²), -3 (k), -2 (constant)

Root at k = -1:

Bring down 1

Multiply by -1: -1

Add to next coefficient: 0 + (-1) = -1

Multiply by -1: 1

Add to next coefficient: -3 + 1 = -2

Multiply by -1: 2

Add to last coefficient: -2 + 2 = 0. Perfect.

Therefore, the cubic factors as (k + 1)(k² - k - 2). Now factor the quadratic:

k² - k - 2 = (k - 2)(k + 1). Therefore, the cubic factors as (k + 1)^2(k - 2). Therefore, roots are k = -1 (double root) and k = 2.

Since k > 0, we discard k = -1. Therefore, the critical point is at k = 2.

So, the minimum occurs at k = 2? Let me check if this is a minimum. Let me check the second derivative or test intervals.

Alternatively, since there's only one critical point for k > 0, and as k approaches 0+ and infinity, f(k) tends to infinity, so this critical point must be the minimum.

Let me confirm by testing values around k=2.

For example, take k=1:

f’(1) = 16 - 48/1 - 32/1 = 16 - 48 -32 = -64 < 0

At k=2:

f’(2) = 0

At k=3:

f’(3) = 16 - 48/9 - 32/27 ≈ 16 - 5.333 - 1.185 ≈ 16 - 6.518 ≈ 9.482 > 0

Therefore, derivative goes from negative to positive at k=2, so it's a minimum. Therefore, the minimum occurs at k=2.

Therefore, the minimal value is f(2). Let me compute f(2):

Earlier, we had f(k) = 8*(2k + 5 + 6/k + 2/k²). Let me compute that at k=2:

2k = 4, 5 remains, 6/k = 3, 2/k² = 0.5. So total inside the brackets: 4 + 5 + 3 + 0.5 = 12.5. Then multiply by 8: 8 * 12.5 = 100. Alternatively, compute using the original expression:

Earlier, f(k) = 8*(2k³ + 5k² + 6k + 2)/k². At k=2:

Numerator: 2*(8) + 5*(4) + 6*(2) + 2 = 16 + 20 + 12 + 2 = 50. Then divide by k² = 4: 50 / 4 = 12.5. Multiply by 8: 100. So yes, 100.

Therefore, the minimal value is 100? Wait, but hold on. Wait, this was under the assumption that a = b. But is this the actual minimum for the original problem? Because we assumed a = b. Maybe there's a lower value when a ≠ b?

Hmm, need to check if the assumption a = b is valid. Since the original problem is symmetric in a and b, it's common that the minimum occurs when a = b. But to be thorough, maybe I should check if this is indeed the case.

Alternatively, maybe we can use Lagrange multipliers for the original problem without assuming a = b. Let me try that approach.

Let me denote the original expression:

$$
\frac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc}
$$

Let me denote variables x = a, y = b, z = c. Let me denote the function as:

f(a, b, c) = [(a + b + c)( (a + b)^2 + (a + b + 4c)^2 )] / (abc)

We need to find the minimum of f(a, b, c) for a, b, c > 0.

To use Lagrange multipliers, maybe set up partial derivatives. However, this might be complex. Alternatively, maybe use substitution.

Alternatively, let me try to use substitution variables. Let me set variables as follows:

Let me denote s = a + b, t = c. Then, as before, the numerator becomes (s + t)[s² + (s + 4t)^2]. The denominator is abc. However, abc is not directly expressible in terms of s and t unless we have more relations. However, if we assume that a = b, then s = 2a, and abc = a^2 t. But if we don't assume a = b, then we need to relate a and b. However, since the problem is symmetric in a and b, perhaps the minimum occurs when a = b, so our previous result of 100 is valid. However, to confirm, let me try to see if there's a way to get a lower value.

Alternatively, maybe use AM-GM inequality. Let me think.

First, let me note that the expression is homogeneous. Let me check the degrees. Let me see:

The numerator: (a + b + c) is degree 1. The term [(a + b)^2 + (a + b + 4c)^2] is degree 2. So total numerator is degree 1 + 2 = 3. The denominator is abc, which is degree 3. Therefore, the entire expression is homogeneous of degree 0. Therefore, we can set one variable to 1 to simplify. For example, set c = 1, then find the minimum in terms of a and b. Or set a + b + c = 1, but maybe setting c = 1 is easier.

Alternatively, since it's homogeneous, we can set c = 1, then find the minimum over a and b. Let me try that.

Let me set c = 1. Then the expression becomes:

[(a + b + 1)[(a + b)^2 + (a + b + 4)^2]] / (ab * 1) = [(a + b + 1)( (a + b)^2 + (a + b + 4)^2 )] / (ab)

Let me denote s = a + b. Then, since a and b are positive, s > 0. Also, ab ≤ (s²)/4 by AM-GM. Therefore, the denominator ab is maximized when a = b, so to minimize the entire expression, since denominator is in the denominator, we need to maximize ab. Therefore, if we fix s, then the minimal value of the expression occurs when ab is maximized, i.e., when a = b. Therefore, again, this suggests that the minimum occurs when a = b. Therefore, our previous assumption is valid. Therefore, the minimal value is indeed 100.

But let me check with c = 1, a = b. Let me set a = b = t, c = 1. Then:

Numerator: (2t + 1)[(2t)^2 + (2t + 4)^2] = (2t + 1)[4t² + (4t² + 16t + 16)] = (2t + 1)[8t² + 16t + 16]

Denominator: t * t * 1 = t²

Therefore, the expression is [(2t + 1)(8t² + 16t + 16)] / t². Let me compute this for t such that k = t / c = t / 1 = t. Wait, earlier we had k = t / c, but here c = 1, so k = t. Wait, earlier when we set a = b = t, c = something, but here c = 1. Wait, in our previous substitution, we set k = t / c, so if c = 1, then k = t. Therefore, the expression becomes f(k) = 8*(2k + 5 + 6/k + 2/k²) when c = 1? Wait, no, earlier when we set a = b = t, c = something, then we set k = t / c. But here, if we set c = 1, then k = t / 1 = t. Therefore, the expression is f(k) = 8*(2k + 5 + 6/k + 2/k²). Wait, but earlier when we set k = 2, we had f(k) = 100. Let me check with c = 1, a = b = t = 2. Then:

Numerator: (2*2 + 1)[(2*2)^2 + (2*2 + 4)^2] = (5)[16 + (8)^2] = 5*(16 + 64) = 5*80 = 400

Denominator: (2)^2 * 1 = 4

Therefore, the expression is 400 / 4 = 100. Correct. So that's consistent.

But let me check if there's a lower value when a ≠ b. Let me try some numbers. For example, take c = 1, a = 1, b approaching 0. Then the expression becomes:

Numerator: (1 + 0 + 1)[(1 + 0)^2 + (1 + 0 + 4*1)^2] = 2[1 + 25] = 2*26 = 52

Denominator: 1*0*1 = 0. Wait, but b approaches 0, so denominator approaches 0, so the expression approaches infinity. Not helpful.

Another test: take a = 1, b = 1, c = 1. Then:

Numerator: (1 + 1 + 1)[(1 + 1)^2 + (1 + 1 + 4*1)^2] = 3[4 + 36] = 3*40 = 120

Denominator: 1*1*1 = 1. So expression is 120. Which is higher than 100. So worse.

Another test: take a = b = 2, c = 1 gives 100. What if take a = b = 1, c = something else? Let me try. Suppose a = b = 1, then let me find c that minimizes the expression. Let me compute the expression as a function of c:

Numerator: (1 + 1 + c)[(2)^2 + (2 + 4c)^2] = (2 + c)[4 + (2 + 4c)^2]

Denominator: 1*1*c = c

So expression is [(2 + c)(4 + (2 + 4c)^2)] / c

Let me compute this for c = 1: [(3)(4 + 36)] / 1 = 3*40 = 120, as before.

What if c = 0.5?

Numerator: (2 + 0.5)[4 + (2 + 2)^2] = 2.5[4 + 16] = 2.5*20 = 50

Denominator: 0.5

Expression: 50 / 0.5 = 100. Wait, same as before? Wait, that's interesting. Wait, if a = b = 1, c = 0.5, then the expression is 100? Let me check again:

Wait, a = 1, b = 1, c = 0.5:

Numerator: (1 + 1 + 0.5)[(1 + 1)^2 + (1 + 1 + 4*0.5)^2] = (2.5)[(2)^2 + (2 + 2)^2] = 2.5[4 + 16] = 2.5*20 = 50

Denominator: 1*1*0.5 = 0.5

So 50 / 0.5 = 100. So same value. Interesting. So even when a = b = 1, c = 0.5, we get 100. But according to our previous substitution, when we set a = b = t, c = something, and set k = t / c, then k = t / c. In this case, t = 1, c = 0.5, so k = 2. Which is the same critical point. Therefore, this is consistent. Therefore, even if we take different values of a, b, c as long as k = t / c = 2, we get the same value. Therefore, the minimum is indeed 100.

But let me check another case where a ≠ b. Let me try a = 3, b = 1, c = ?

Wait, but how to choose c? Maybe set k such that... Wait, maybe not. Let me try to set a = 3, b = 1, and find c that minimizes the expression. But this might be complicated. Alternatively, let me try to use Lagrange multipliers.

Let me try that. Let me consider the original function:

f(a, b, c) = [(a + b + c)( (a + b)^2 + (a + b + 4c)^2 )] / (abc)

Let me take natural logarithm to make differentiation easier. Let me denote:

Let me set F(a, b, c) = ln f(a, b, c) = ln(a + b + c) + ln[(a + b)^2 + (a + b + 4c)^2] - ln a - ln b - ln c

Then, to find critical points, take partial derivatives with respect to a, b, c and set them to zero.

Compute ∂F/∂a:

First term: derivative of ln(a + b + c) is 1/(a + b + c)

Second term: derivative of ln[(a + b)^2 + (a + b + 4c)^2] with respect to a:

Let me denote u = (a + b)^2 + (a + b + 4c)^2. Then derivative is [2(a + b) + 2(a + b + 4c)(1)] / u = [2(a + b) + 2(a + b + 4c)] / u = [4(a + b) + 8c]/u

Third term: derivative of -ln a is -1/a

Fourth term: derivative of -ln b is 0

Fifth term: derivative of -ln c is 0

Therefore, ∂F/∂a = 1/(a + b + c) + [4(a + b) + 8c]/[(a + b)^2 + (a + b + 4c)^2] - 1/a = 0

Similarly, ∂F/∂b will be the same as ∂F/∂a because of symmetry in a and b. Therefore, ∂F/∂b = same expression with a and b swapped, but since the expression is symmetric, it will be same as ∂F/∂a. Therefore, we can focus on ∂F/∂a and ∂F/∂c.

Compute ∂F/∂c:

First term: derivative of ln(a + b + c) is 1/(a + b + c)

Second term: derivative of ln[(a + b)^2 + (a + b + 4c)^2] with respect to c:

Again, u = (a + b)^2 + (a + b + 4c)^2. Then derivative is [2(a + b + 4c)(4)] / u = 8(a + b + 4c)/u

Third term: derivative of -ln a is 0

Fourth term: derivative of -ln b is 0

Fifth term: derivative of -ln c is -1/c

Therefore, ∂F/∂c = 1/(a + b + c) + 8(a + b + 4c)/[(a + b)^2 + (a + b + 4c)^2] - 1/c = 0

So now, we have three equations:

1. 1/(a + b + c) + [4(a + b) + 8c]/[(a + b)^2 + (a + b + 4c)^2] - 1/a = 0

2. Same as equation 1 with a and b swapped, but since symmetric, we can ignore.

3. 1/(a + b + c) + 8(a + b + 4c)/[(a + b)^2 + (a + b + 4c)^2] - 1/c = 0

Let me denote S = a + b, T = a + b + 4c. Then note that T = S + 4c. Also, note that the denominator in the second term of equation 1 is S² + T². Let me rewrite equations:

Equation 1:

1/(S + c) + [4S + 8c]/(S² + T²) - 1/a = 0

Equation 3:

1/(S + c) + 8T/(S² + T²) - 1/c = 0

Also, note that T = S + 4c.

Let me try to use the previous assumption that a = b. Let me set a = b, so S = 2a, and T = 2a + 4c. Let me check if equations are satisfied.

Let me denote a = b, so S = 2a, T = 2a + 4c.

Equation 1:

1/(2a + c) + [4*(2a) + 8c]/[(2a)^2 + (2a + 4c)^2] - 1/a = 0

Compute numerator of the second term: 8a + 8c = 8(a + c)

Denominator: 4a² + (2a + 4c)^2 = 4a² + 4a² + 16ac + 16c² = 8a² + 16ac + 16c² = 8(a² + 2ac + 2c²)

Therefore, second term is 8(a + c)/[8(a² + 2ac + 2c²)] = (a + c)/(a² + 2ac + 2c²)

Therefore, equation 1 becomes:

1/(2a + c) + (a + c)/(a² + 2ac + 2c²) - 1/a = 0

Similarly, equation 3:

1/(2a + c) + 8*(2a + 4c)/[ (2a)^2 + (2a + 4c)^2 ] - 1/c = 0

Compute numerator of second term: 8*(2a + 4c) = 16a + 32c

Denominator same as before: 8(a² + 2ac + 2c²)

Therefore, second term is (16a + 32c)/[8(a² + 2ac + 2c²)] = 2(a + 2c)/(a² + 2ac + 2c²)

Therefore, equation 3 becomes:

1/(2a + c) + 2(a + 2c)/(a² + 2ac + 2c²) - 1/c = 0

Now, let me check if with a = b and k = a/c = 2, which was our previous critical point. Let me set k = a/c = 2 => a = 2c. Let me substitute a = 2c into equations.

First, let me compute equation 1:

Let a = 2c. Then S = 2a = 4c, T = 2a + 4c = 8c.

Compute 1/(2a + c) = 1/(4c + c) = 1/(5c)

Second term: (a + c)/(a² + 2ac + 2c²) = (2c + c)/[(4c²) + 2*(2c)*c + 2c²] = 3c / [4c² + 4c² + 2c²] = 3c / 10c² = 3/(10c)

Third term: -1/a = -1/(2c)

Therefore, equation 1 becomes:

1/(5c) + 3/(10c) - 1/(2c) = [2/(10c) + 3/(10c) - 5/(10c)] = 0. Correct. So equation 1 is satisfied.

Now check equation 3:

1/(2a + c) = 1/(5c)

Second term: 2(a + 2c)/(a² + 2ac + 2c²) = 2(2c + 2c)/[4c² + 4c² + 2c²] = 2*(4c)/10c² = 8c / 10c² = 4/(5c)

Third term: -1/c

Therefore, equation 3 becomes:

1/(5c) + 4/(5c) - 1/c = (5/(5c)) - 1/c = 1/c - 1/c = 0. Correct. Therefore, equations are satisfied when a = 2c, i.e., k = 2. Therefore, this confirms that our previous result is indeed a critical point.

Therefore, since we have checked that with a = b and c = something, we get the critical point, and also that the function tends to infinity as variables approach boundaries, this critical point is indeed the global minimum. Therefore, the minimal value is 100.

But just to be thorough, let me check if there are other critical points when a ≠ b. Suppose there is a case where a ≠ b. However, given the symmetry in a and b, any critical point with a ≠ b would have a corresponding one with a and b swapped, but since the function is symmetric, they would give the same value. However, without loss of generality, we can assume a = b due to symmetry, so the minimum occurs at a = b. Therefore, our answer of 100 is correct.

Therefore, the minimum value is \boxed{100}.

**Final Answer**
\boxed{100}

To find the minimum value of the expression:

$$
\frac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc}
$$

where $ a, b, c > 0 $, we begin by analyzing the structure of the expression. It is a rational function involving symmetric terms in $ a $ and $ b $, and a linear term in $ c $. The numerator is a product of a linear expression and a sum of squares, while the denominator is the product of the three variables.

---

### Step 1: Symmetry and Substitution

The expression is symmetric in $ a $ and $ b $, suggesting that the minimum may occur when $ a = b $. Let us assume $ a = b = t $, and let $ c = y $. Then:

- $ a + b = 2t $
- $ a + b + c = 2t + y $
- $ (a + b)^2 = (2t)^2 = 4t^2 $
- $ (a + b + 4c)^2 = (2t + 4y)^2 = 4t^2 + 16ty + 16y^2 $

So the numerator becomes:

$$
(2t + y)\left[4t^2 + (2t + 4y)^2\right] = (2t + y)\left[4t^2 + 4t^2 + 16ty + 16y^2\right] = (2t + y)(8t^2 + 16ty + 16y^2)
$$

The denominator is:

$$
abc = t \cdot t \cdot y = t^2 y
$$

Thus, the expression becomes:

$$
\frac{(2t + y)(8t^2 + 16ty + 16y^2)}{t^2 y}
$$

---

### Step 2: Introduce a Ratio

Let us define $ k = \frac{t}{y} $, so $ t = ky $. Substituting into the expression:

- $ 2t + y = 2ky + y = y(2k + 1) $
- $ t^2 = k^2 y^2 $
- $ 8t^2 + 16ty + 16y^2 = 8k^2 y^2 + 16k y^2 + 16y^2 = y^2(8k^2 + 16k + 16) $

So the expression becomes:

$$
\frac{y(2k + 1) \cdot y^2(8k^2 + 16k + 16)}{k^2 y^3} = \frac{(2k + 1)(8k^2 + 16k + 16)}{k^2}
$$

Simplifying:

$$
f(k) = \frac{(2k + 1)(8k^2 + 16k + 16)}{k^2}
$$

---

### Step 3: Minimize the Function

We now minimize the function:

$$
f(k) = \frac{(2k + 1)(8k^2 + 16k + 16)}{k^2}
$$

Expanding the numerator:

$$
(2k + 1)(8k^2 + 16k + 16) = 16k^3 + 32k^2 + 32k + 8k^2 + 16k + 16 = 16k^3 + 40k^2 + 48k + 16
$$

So:

$$
f(k) = \frac{16k^3 + 40k^2 + 48k + 16}{k^2} = 16k + 40 + \frac{48}{k} + \frac{16}{k^2}
$$

To find the minimum, take the derivative:

$$
f'(k) = 16 - \frac{48}{k^2} - \frac{32}{k^3}
$$

Set $ f'(k) = 0 $:

$$
16 - \frac{48}{k^2} - \frac{32}{k^3} = 0
$$

Multiply through by $ k^3 $:

$$
16k^3 - 48k - 32 = 0
$$

Divide by 16:

$$
k^3 - 3k - 2 = 0
$$

This cubic factors as:

$$
(k + 1)^2(k - 2) = 0
$$

So the only positive root is $ k = 2 $.

---

### Step 4: Evaluate the Minimum

Substitute $ k = 2 $ into the expression:

$$
f(2) = \frac{(2 \cdot 2 + 1)(8 \cdot 2^2 + 16 \cdot 2 + 16)}{2^2} = \frac{5 \cdot (32 + 32 + 16)}{4} = \frac{5 \cdot 80}{4} = \frac{400}{4} = 100
$$

---

### Step 5: Confirm Optimality

Since the function is symmetric in $ a $ and $ b $, and the expression is homogeneous of degree 0, the minimum occurs when $ a = b $, and the value is minimized at $ k = 2 $. Further, as $ k \to 0^+ $ or $ k \to \infty $, the function tends to infinity, confirming that the critical point at $ k = 2 $ is indeed the global minimum.

---

### Final Answer

$$
\boxed{100}
$$

[Assistant]
Explanation:
Okay, so I need to find the minimum value of this expression:

$$
\frac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc}
$$

where $ a $, $ b $, and $ c $ are positive real numbers. Hmm, let me think about how to approach this. It looks like an optimization problem with three variables. Maybe I can use calculus, like taking partial derivatives and setting them to zero? But that might get complicated with three variables. Alternatively, maybe there's a way to apply inequalities like AM-GM or Cauchy-Schwarz? Let me try to simplify the expression first.

First, let me expand the numerator. Let me denote $ S = a + b + c $, but maybe that's not helpful yet. Let me look at the terms inside the brackets:

$$
(a + b)^2 + (a + b + 4c)^2
$$

Let me compute each term:

First term: $ (a + b)^2 = a^2 + 2ab + b^2 $

Second term: $ (a + b + 4c)^2 = (a + b)^2 + 8c(a + b) + 16c^2 $

So adding them together:

$$
(a + b)^2 + (a + b + 4c)^2 = [a^2 + 2ab + b^2] + [a^2 + 2ab + b^2 + 8c(a + b) + 16c^2]
$$

Combine like terms:

- $ a^2 + a^2 = 2a^2 $
- $ 2ab + 2ab = 4ab $
- $ b^2 + b^2 = 2b^2 $
- Then the other terms: $ 8c(a + b) + 16c^2 $

So altogether:

$$
2a^2 + 4ab + 2b^2 + 8c(a + b) + 16c^2
$$

Hmm, maybe factor some terms? Let me see:

First, notice that $ 2a^2 + 4ab + 2b^2 = 2(a^2 + 2ab + b^2) = 2(a + b)^2 $. So that simplifies part of it. Then the rest is $ 8c(a + b) + 16c^2 $. So the entire numerator becomes:

$$
(a + b + c) \times [2(a + b)^2 + 8c(a + b) + 16c^2]
$$

Let me write that as:

$$
(a + b + c) \times [2(a + b)^2 + 8c(a + b) + 16c^2]
$$

Hmm, maybe factor out a 2 from the bracketed terms? Let me check:

Wait, 2(a + b)^2 + 8c(a + b) + 16c^2. Let me factor 2 from the first two terms:

2[(a + b)^2 + 4c(a + b)] + 16c^2. Hmm, not sure if that helps. Alternatively, maybe set variables to simplify. Let me think.

Let me denote $ x = a + b $, and $ y = c $. Then the expression becomes:

Numerator: $ (x + y)[2x^2 + 8y x + 16y^2] $

Denominator: $ abc $. But $ a $ and $ b $ are still variables. However, since $ x = a + b $, maybe I can express $ ab $ in terms of $ x $ and another variable? For example, if I fix $ x = a + b $, then $ ab $ is maximized when $ a = b $, but since we are dealing with a ratio, maybe we need to find the minimum of the entire expression. Wait, but the denominator is $ abc $, so if I fix $ x = a + b $, then $ ab $ is related to $ a $ and $ b $, but $ c $ is separate. Maybe I need to consider variables substitution.

Alternatively, maybe use symmetry or set variables in terms of each other. Let me think. Let me try to assume some relation between a, b, c. For example, maybe set a = b? Since the expression is symmetric in a and b? Let me check.

Looking at the original expression:

The numerator has (a + b + c) multiplied by [(a + b)^2 + (a + b + 4c)^2]. The denominator is abc. Since the expression is symmetric in a and b, maybe the minimum occurs when a = b? Let me assume that a = b. Let me set a = b = t, and then express everything in terms of t and c. Let me try that.

Let me set a = b = t. Then:

Numerator:

First, a + b + c = 2t + c

Then, (a + b)^2 + (a + b + 4c)^2 = (2t)^2 + (2t + 4c)^2 = 4t² + (2t + 4c)²

Compute that:

(2t + 4c)² = 4t² + 16tc + 16c²

So total numerator inside the brackets: 4t² + 4t² + 16tc + 16c² = 8t² + 16tc + 16c²

Therefore, the entire numerator is (2t + c)(8t² + 16tc + 16c²)

Denominator is abc = t * t * c = t²c

So the entire expression becomes:

[ (2t + c)(8t² + 16tc + 16c²) ] / (t²c)

Let me simplify this expression. Let me factor numerator:

First, note that 8t² + 16tc + 16c² = 8(t² + 2tc + 2c²). Wait, let me check:

Wait 8t² + 16tc + 16c² = 8t² + 16tc + 16c². Let me factor 8 from the first term and 16 from the others? Not sure. Alternatively, factor 8:

Wait 8t² + 16tc + 16c² = 8(t² + 2tc) + 16c²? Not helpful. Alternatively, maybe factor 8t² + 16tc + 16c² = 8(t² + 2tc + 2c²). Wait, 8(t² + 2tc + 2c²) would be 8t² + 16tc + 16c². Yes, exactly. So that's correct.

So numerator becomes (2t + c) * 8(t² + 2tc + 2c²) = 8(2t + c)(t² + 2tc + 2c²)

Denominator is t²c. Therefore, the entire expression is:

8(2t + c)(t² + 2tc + 2c²) / (t²c)

Hmm, maybe simplify this expression. Let me denote variables in terms of ratios. Let me set k = t/c. Since t and c are positive, k is positive. Let me set k = t/c, so t = kc. Then substitute into the expression.

So t = kc. Then:

First, 2t + c = 2kc + c = c(2k + 1)

Next, t² + 2tc + 2c² = (k²c²) + 2(kc)c + 2c² = c²(k² + 2k + 2)

Therefore, numerator becomes 8 * c(2k + 1) * c²(k² + 2k + 2) = 8c³(2k + 1)(k² + 2k + 2)

Denominator is t²c = (k²c²)c = k²c³

Therefore, the entire expression becomes:

[8c³(2k + 1)(k² + 2k + 2)] / [k²c³] = 8(2k + 1)(k² + 2k + 2)/k²

So now, the problem reduces to minimizing the function f(k) = 8(2k + 1)(k² + 2k + 2)/k² where k > 0.

Okay, so now it's a single-variable calculus problem. Let me compute f(k):

First, expand the numerator:

(2k + 1)(k² + 2k + 2). Let me compute that:

Multiply term by term:

First, 2k*(k²) = 2k³

2k*(2k) = 4k²

2k*(2) = 4k

Then 1*(k²) = k²

1*(2k) = 2k

1*(2) = 2

So adding all together:

2k³ + 4k² + 4k + k² + 2k + 2 = 2k³ + (4k² + k²) + (4k + 2k) + 2 = 2k³ + 5k² + 6k + 2

Therefore, f(k) = 8*(2k³ + 5k² + 6k + 2)/k² = 8*(2k³/k² + 5k²/k² + 6k/k² + 2/k²) = 8*(2k + 5 + 6/k + 2/k²)

So f(k) = 8*(2k + 5 + 6/k + 2/k²)

Alternatively, maybe keep it as 8*(2k³ + 5k² + 6k + 2)/k². Either way, to find the minimum, take derivative with respect to k and set to zero.

Let me compute f(k) = 8*(2k³ + 5k² + 6k + 2)/k². Let me denote f(k) = 8*(2k³ + 5k² + 6k + 2)/k². Let me compute derivative f’(k):

First, let me write f(k) as 8*(2k³ + 5k² + 6k + 2)*k^{-2}

So derivative f’(k) = 8*[ derivative of (2k³ + 5k² + 6k + 2)*k^{-2} + (2k³ + 5k² + 6k + 2)* derivative of k^{-2} ]

Wait, actually, using product rule:

Let me denote u = 2k³ + 5k² + 6k + 2 and v = k^{-2}

Then f(k) = 8uv, so f’(k) = 8(u’v + uv’)

Compute u’:

u’ = 6k² + 10k + 6

v’ = -2k^{-3}

Therefore:

f’(k) = 8[ (6k² + 10k + 6)(k^{-2}) + (2k³ + 5k² + 6k + 2)(-2k^{-3}) ]

Let me compute each term:

First term: (6k² + 10k + 6)/k² = 6 + 10/k + 6/k²

Second term: -2(2k³ + 5k² + 6k + 2)/k³ = -2*(2 + 5/k + 6/k² + 2/k³)

Therefore, combining:

f’(k) = 8[ (6 + 10/k + 6/k²) - 2*(2 + 5/k + 6/k² + 2/k³) ]

Let me compute inside the brackets:

First expand the second part:

-2*(2 + 5/k + 6/k² + 2/k³) = -4 -10/k -12/k² -4/k³

Now add the first part:

(6 + 10/k + 6/k²) + (-4 -10/k -12/k² -4/k³) =

Compute term by term:

6 - 4 = 2

10/k -10/k = 0

6/k² -12/k² = -6/k²

Then -4/k³

So overall: 2 - 6/k² - 4/k³

Therefore, f’(k) = 8*(2 - 6/k² - 4/k³) = 16 - 48/k² - 32/k³

Set derivative equal to zero:

16 - 48/k² - 32/k³ = 0

Multiply both sides by k³ to eliminate denominators:

16k³ - 48k - 32 = 0

Simplify:

Divide both sides by 16:

k³ - 3k - 2 = 0

So we have the equation k³ - 3k - 2 = 0. Let me solve this cubic equation.

Let me try rational roots. Possible rational roots are ±1, ±2.

Test k=1: 1 - 3 - 2 = -4 ≠ 0

k=-1: -1 + 3 - 2 = 0. Oh, k=-1 is a root. Therefore, we can factor (k + 1) from the cubic.

Using polynomial division or synthetic division:

Divide k³ - 3k - 2 by (k + 1). Let me do synthetic division:

Coefficients: 1 (k³), 0 (k²), -3 (k), -2 (constant)

Root at k = -1:

Bring down 1

Multiply by -1: -1

Add to next coefficient: 0 + (-1) = -1

Multiply by -1: 1

Add to next coefficient: -3 + 1 = -2

Multiply by -1: 2

Add to last coefficient: -2 + 2 = 0. Perfect.

Therefore, the cubic factors as (k + 1)(k² - k - 2). Now factor the quadratic:

k² - k - 2 = (k - 2)(k + 1). Therefore, the cubic factors as (k + 1)^2(k - 2). Therefore, roots are k = -1 (double root) and k = 2.

Since k > 0, we discard k = -1. Therefore, the critical point is at k = 2.

So, the minimum occurs at k = 2? Let me check if this is a minimum. Let me check the second derivative or test intervals.

Alternatively, since there's only one critical point for k > 0, and as k approaches 0+ and infinity, f(k) tends to infinity, so this critical point must be the minimum.

Let me confirm by testing values around k=2.

For example, take k=1:

f’(1) = 16 - 48/1 - 32/1 = 16 - 48 -32 = -64 < 0

At k=2:

f’(2) = 0

At k=3:

f’(3) = 16 - 48/9 - 32/27 ≈ 16 - 5.333 - 1.185 ≈ 16 - 6.518 ≈ 9.482 > 0

Therefore, derivative goes from negative to positive at k=2, so it's a minimum. Therefore, the minimum occurs at k=2.

Therefore, the minimal value is f(2). Let me compute f(2):

Earlier, we had f(k) = 8*(2k + 5 + 6/k + 2/k²). Let me compute that at k=2:

2k = 4, 5 remains, 6/k = 3, 2/k² = 0.5. So total inside the brackets: 4 + 5 + 3 + 0.5 = 12.5. Then multiply by 8: 8 * 12.5 = 100. Alternatively, compute using the original expression:

Earlier, f(k) = 8*(2k³ + 5k² + 6k + 2)/k². At k=2:

Numerator: 2*(8) + 5*(4) + 6*(2) + 2 = 16 + 20 + 12 + 2 = 50. Then divide by k² = 4: 50 / 4 = 12.5. Multiply by 8: 100. So yes, 100.

Therefore, the minimal value is 100? Wait, but hold on. Wait, this was under the assumption that a = b. But is this the actual minimum for the original problem? Because we assumed a = b. Maybe there's a lower value when a ≠ b?

Hmm, need to check if the assumption a = b is valid. Since the original problem is symmetric in a and b, it's common that the minimum occurs when a = b. But to be thorough, maybe I should check if this is indeed the case.

Alternatively, maybe we can use Lagrange multipliers for the original problem without assuming a = b. Let me try that approach.

Let me denote the original expression:

$$
\frac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc}
$$

Let me denote variables x = a, y = b, z = c. Let me denote the function as:

f(a, b, c) = [(a + b + c)( (a + b)^2 + (a + b + 4c)^2 )] / (abc)

We need to find the minimum of f(a, b, c) for a, b, c > 0.

To use Lagrange multipliers, maybe set up partial derivatives. However, this might be complex. Alternatively, maybe use substitution.

Alternatively, let me try to use substitution variables. Let me set variables as follows:

Let me denote s = a + b, t = c. Then, as before, the numerator becomes (s + t)[s² + (s + 4t)^2]. The denominator is abc. However, abc is not directly expressible in terms of s and t unless we have more relations. However, if we assume that a = b, then s = 2a, and abc = a^2 t. But if we don't assume a = b, then we need to relate a and b. However, since the problem is symmetric in a and b, perhaps the minimum occurs when a = b, so our previous result of 100 is valid. However, to confirm, let me try to see if there's a way to get a lower value.

Alternatively, maybe use AM-GM inequality. Let me think.

First, let me note that the expression is homogeneous. Let me check the degrees. Let me see:

The numerator: (a + b + c) is degree 1. The term [(a + b)^2 + (a + b + 4c)^2] is degree 2. So total numerator is degree 1 + 2 = 3. The denominator is abc, which is degree 3. Therefore, the entire expression is homogeneous of degree 0. Therefore, we can set one variable to 1 to simplify. For example, set c = 1, then find the minimum in terms of a and b. Or set a + b + c = 1, but maybe setting c = 1 is easier.

Alternatively, since it's homogeneous, we can set c = 1, then find the minimum over a and b. Let me try that.

Let me set c = 1. Then the expression becomes:

[(a + b + 1)[(a + b)^2 + (a + b + 4)^2]] / (ab * 1) = [(a + b + 1)( (a + b)^2 + (a + b + 4)^2 )] / (ab)

Let me denote s = a + b. Then, since a and b are positive, s > 0. Also, ab ≤ (s²)/4 by AM-GM. Therefore, the denominator ab is maximized when a = b, so to minimize the entire expression, since denominator is in the denominator, we need to maximize ab. Therefore, if we fix s, then the minimal value of the expression occurs when ab is maximized, i.e., when a = b. Therefore, again, this suggests that the minimum occurs when a = b. Therefore, our previous assumption is valid. Therefore, the minimal value is indeed 100.

But let me check with c = 1, a = b. Let me set a = b = t, c = 1. Then:

Numerator: (2t + 1)[(2t)^2 + (2t + 4)^2] = (2t + 1)[4t² + (4t² + 16t + 16)] = (2t + 1)[8t² + 16t + 16]

Denominator: t * t * 1 = t²

Therefore, the expression is [(2t + 1)(8t² + 16t + 16)] / t². Let me compute this for t such that k = t / c = t / 1 = t. Wait, earlier we had k = t / c, but here c = 1, so k = t. Wait, earlier when we set a = b = t, c = something, but here c = 1. Wait, in our previous substitution, we set k = t / c, so if c = 1, then k = t. Therefore, the expression becomes f(k) = 8*(2k + 5 + 6/k + 2/k²) when c = 1? Wait, no, earlier when we set a = b = t, c = something, then we set k = t / c. But here, if we set c = 1, then k = t / 1 = t. Therefore, the expression is f(k) = 8*(2k + 5 + 6/k + 2/k²). Wait, but earlier when we set k = 2, we had f(k) = 100. Let me check with c = 1, a = b = t = 2. Then:

Numerator: (2*2 + 1)[(2*2)^2 + (2*2 + 4)^2] = (5)[16 + (8)^2] = 5*(16 + 64) = 5*80 = 400

Denominator: (2)^2 * 1 = 4

Therefore, the expression is 400 / 4 = 100. Correct. So that's consistent.

But let me check if there's a lower value when a ≠ b. Let me try some numbers. For example, take c = 1, a = 1, b approaching 0. Then the expression becomes:

Numerator: (1 + 0 + 1)[(1 + 0)^2 + (1 + 0 + 4*1)^2] = 2[1 + 25] = 2*26 = 52

Denominator: 1*0*1 = 0. Wait, but b approaches 0, so denominator approaches 0, so the expression approaches infinity. Not helpful.

Another test: take a = 1, b = 1, c = 1. Then:

Numerator: (1 + 1 + 1)[(1 + 1)^2 + (1 + 1 + 4*1)^2] = 3[4 + 36] = 3*40 = 120

Denominator: 1*1*1 = 1. So expression is 120. Which is higher than 100. So worse.

Another test: take a = b = 2, c = 1 gives 100. What if take a = b = 1, c = something else? Let me try. Suppose a = b = 1, then let me find c that minimizes the expression. Let me compute the expression as a function of c:

Numerator: (1 + 1 + c)[(2)^2 + (2 + 4c)^2] = (2 + c)[4 + (2 + 4c)^2]

Denominator: 1*1*c = c

So expression is [(2 + c)(4 + (2 + 4c)^2)] / c

Let me compute this for c = 1: [(3)(4 + 36)] / 1 = 3*40 = 120, as before.

What if c = 0.5?

Numerator: (2 + 0.5)[4 + (2 + 2)^2] = 2.5[4 + 16] = 2.5*20 = 50

Denominator: 0.5

Expression: 50 / 0.5 = 100. Wait, same as before? Wait, that's interesting. Wait, if a = b = 1, c = 0.5, then the expression is 100? Let me check again:

Wait, a = 1, b = 1, c = 0.5:

Numerator: (1 + 1 + 0.5)[(1 + 1)^2 + (1 + 1 + 4*0.5)^2] = (2.5)[(2)^2 + (2 + 2)^2] = 2.5[4 + 16] = 2.5*20 = 50

Denominator: 1*1*0.5 = 0.5

So 50 / 0.5 = 100. So same value. Interesting. So even when a = b = 1, c = 0.5, we get 100. But according to our previous substitution, when we set a = b = t, c = something, and set k = t / c, then k = t / c. In this case, t = 1, c = 0.5, so k = 2. Which is the same critical point. Therefore, this is consistent. Therefore, even if we take different values of a, b, c as long as k = t / c = 2, we get the same value. Therefore, the minimum is indeed 100.

But let me check another case where a ≠ b. Let me try a = 3, b = 1, c = ?

Wait, but how to choose c? Maybe set k such that... Wait, maybe not. Let me try to set a = 3, b = 1, and find c that minimizes the expression. But this might be complicated. Alternatively, let me try to use Lagrange multipliers.

Let me try that. Let me consider the original function:

f(a, b, c) = [(a + b + c)( (a + b)^2 + (a + b + 4c)^2 )] / (abc)

Let me take natural logarithm to make differentiation easier. Let me denote:

Let me set F(a, b, c) = ln f(a, b, c) = ln(a + b + c) + ln[(a + b)^2 + (a + b + 4c)^2] - ln a - ln b - ln c

Then, to find critical points, take partial derivatives with respect to a, b, c and set them to zero.

Compute ∂F/∂a:

First term: derivative of ln(a + b + c) is 1/(a + b + c)

Second term: derivative of ln[(a + b)^2 + (a + b + 4c)^2] with respect to a:

Let me denote u = (a + b)^2 + (a + b + 4c)^2. Then derivative is [2(a + b) + 2(a + b + 4c)(1)] / u = [2(a + b) + 2(a + b + 4c)] / u = [4(a + b) + 8c]/u

Third term: derivative of -ln a is -1/a

Fourth term: derivative of -ln b is 0

Fifth term: derivative of -ln c is 0

Therefore, ∂F/∂a = 1/(a + b + c) + [4(a + b) + 8c]/[(a + b)^2 + (a + b + 4c)^2] - 1/a = 0

Similarly, ∂F/∂b will be the same as ∂F/∂a because of symmetry in a and b. Therefore, ∂F/∂b = same expression with a and b swapped, but since the expression is symmetric, it will be same as ∂F/∂a. Therefore, we can focus on ∂F/∂a and ∂F/∂c.

Compute ∂F/∂c:

First term: derivative of ln(a + b + c) is 1/(a + b + c)

Second term: derivative of ln[(a + b)^2 + (a + b + 4c)^2] with respect to c:

Again, u = (a + b)^2 + (a + b + 4c)^2. Then derivative is [2(a + b + 4c)(4)] / u = 8(a + b + 4c)/u

Third term: derivative of -ln a is 0

Fourth term: derivative of -ln b is 0

Fifth term: derivative of -ln c is -1/c

Therefore, ∂F/∂c = 1/(a + b + c) + 8(a + b + 4c)/[(a + b)^2 + (a + b + 4c)^2] - 1/c = 0

So now, we have three equations:

1. 1/(a + b + c) + [4(a + b) + 8c]/[(a + b)^2 + (a + b + 4c)^2] - 1/a = 0

2. Same as equation 1 with a and b swapped, but since symmetric, we can ignore.

3. 1/(a + b + c) + 8(a + b + 4c)/[(a + b)^2 + (a + b + 4c)^2] - 1/c = 0

Let me denote S = a + b, T = a + b + 4c. Then note that T = S + 4c. Also, note that the denominator in the second term of equation 1 is S² + T². Let me rewrite equations:

Equation 1:

1/(S + c) + [4S + 8c]/(S² + T²) - 1/a = 0

Equation 3:

1/(S + c) + 8T/(S² + T²) - 1/c = 0

Also, note that T = S + 4c.

Let me try to use the previous assumption that a = b. Let me set a = b, so S = 2a, and T = 2a + 4c. Let me check if equations are satisfied.

Let me denote a = b, so S = 2a, T = 2a + 4c.

Equation 1:

1/(2a + c) + [4*(2a) + 8c]/[(2a)^2 + (2a + 4c)^2] - 1/a = 0

Compute numerator of the second term: 8a + 8c = 8(a + c)

Denominator: 4a² + (2a + 4c)^2 = 4a² + 4a² + 16ac + 16c² = 8a² + 16ac + 16c² = 8(a² + 2ac + 2c²)

Therefore, second term is 8(a + c)/[8(a² + 2ac + 2c²)] = (a + c)/(a² + 2ac + 2c²)

Therefore, equation 1 becomes:

1/(2a + c) + (a + c)/(a² + 2ac + 2c²) - 1/a = 0

Similarly, equation 3:

1/(2a + c) + 8*(2a + 4c)/[ (2a)^2 + (2a + 4c)^2 ] - 1/c = 0

Compute numerator of second term: 8*(2a + 4c) = 16a + 32c

Denominator same as before: 8(a² + 2ac + 2c²)

Therefore, second term is (16a + 32c)/[8(a² + 2ac + 2c²)] = 2(a + 2c)/(a² + 2ac + 2c²)

Therefore, equation 3 becomes:

1/(2a + c) + 2(a + 2c)/(a² + 2ac + 2c²) - 1/c = 0

Now, let me check if with a = b and k = a/c = 2, which was our previous critical point. Let me set k = a/c = 2 => a = 2c. Let me substitute a = 2c into equations.

First, let me compute equation 1:

Let a = 2c. Then S = 2a = 4c, T = 2a + 4c = 8c.

Compute 1/(2a + c) = 1/(4c + c) = 1/(5c)

Second term: (a + c)/(a² + 2ac + 2c²) = (2c + c)/[(4c²) + 2*(2c)*c + 2c²] = 3c / [4c² + 4c² + 2c²] = 3c / 10c² = 3/(10c)

Third term: -1/a = -1/(2c)

Therefore, equation 1 becomes:

1/(5c) + 3/(10c) - 1/(2c) = [2/(10c) + 3/(10c) - 5/(10c)] = 0. Correct. So equation 1 is satisfied.

Now check equation 3:

1/(2a + c) = 1/(5c)

Second term: 2(a + 2c)/(a² + 2ac + 2c²) = 2(2c + 2c)/[4c² + 4c² + 2c²] = 2*(4c)/10c² = 8c / 10c² = 4/(5c)

Third term: -1/c

Therefore, equation 3 becomes:

1/(5c) + 4/(5c) - 1/c = (5/(5c)) - 1/c = 1/c - 1/c = 0. Correct. Therefore, equations are satisfied when a = 2c, i.e., k = 2. Therefore, this confirms that our previous result is indeed a critical point.

Therefore, since we have checked that with a = b and c = something, we get the critical point, and also that the function tends to infinity as variables approach boundaries, this critical point is indeed the global minimum. Therefore, the minimal value is 100.

But just to be thorough, let me check if there are other critical points when a ≠ b. Suppose there is a case where a ≠ b. However, given the symmetry in a and b, any critical point with a ≠ b would have a corresponding one with a and b swapped, but since the function is symmetric, they would give the same value. However, without loss of generality, we can assume a = b due to symmetry, so the minimum occurs at a = b. Therefore, our answer of 100 is correct.

Therefore, the minimum value is \boxed{100}.

**Final Answer**
\boxed{100}

To find the minimum value of the expression:

$$
\frac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc}
$$

where $ a, b, c > 0 $, we begin by analyzing the structure of the expression. It is a rational function involving symmetric terms in $ a $ and $ b $, and a linear term in $ c $. The numerator is a product of a linear expression and a sum of squares, while the denominator is the product of the three variables.

---

### Step 1: Symmetry and Substitution

The expression is symmetric in $ a $ and $ b $, suggesting that the minimum may occur when $ a = b $. Let us assume $ a = b = t $, and let $ c = y $. Then:

- $ a + b = 2t $
- $ a + b + c = 2t + y $
- $ (a + b)^2 = (2t)^2 = 4t^2 $
- $ (a + b + 4c)^2 = (2t + 4y)^2 = 4t^2 + 16ty + 16y^2 $

So the numerator becomes:

$$
(2t + y)\left[4t^2 + (2t + 4y)^2\right] = (2t + y)\left[4t^2 + 4t^2 + 16ty + 16y^2\right] = (2t + y)(8t^2 + 16ty + 16y^2)
$$

The denominator is:

$$
abc = t \cdot t \cdot y = t^2 y
$$

Thus, the expression becomes:

$$
\frac{(2t + y)(8t^2 + 16ty + 16y^2)}{t^2 y}
$$

---

### Step 2: Introduce a Ratio

Let us define $ k = \frac{t}{y} $, so $ t = ky $. Substituting into the expression:

- $ 2t + y = 2ky + y = y(2k + 1) $
- $ t^2 = k^2 y^2 $
- $ 8t^2 + 16ty + 16y^2 = 8k^2 y^2 + 16k y^2 + 16y^2 = y^2(8k^2 + 16k + 16) $

So the expression becomes:

$$
\frac{y(2k + 1) \cdot y^2(8k^2 + 16k + 16)}{k^2 y^3} = \frac{(2k + 1)(8k^2 + 16k + 16)}{k^2}
$$

Simplifying:

$$
f(k) = \frac{(2k + 1)(8k^2 + 16k + 16)}{k^2}
$$

---

### Step 3: Minimize the Function

We now minimize the function:

$$
f(k) = \frac{(2k + 1)(8k^2 + 16k + 16)}{k^2}
$$

Expanding the numerator:

$$
(2k + 1)(8k^2 + 16k + 16) = 16k^3 + 32k^2 + 32k + 8k^2 + 16k + 16 = 16k^3 + 40k^2 + 48k + 16
$$

So:

$$
f(k) = \frac{16k^3 + 40k^2 + 48k + 16}{k^2} = 16k + 40 + \frac{48}{k} + \frac{16}{k^2}
$$

To find the minimum, take the derivative:

$$
f'(k) = 16 - \frac{48}{k^2} - \frac{32}{k^3}
$$

Set $ f'(k) = 0 $:

$$
16 - \frac{48}{k^2} - \frac{32}{k^3} = 0
$$

Multiply through by $ k^3 $:

$$
16k^3 - 48k - 32 = 0
$$

Divide by 16:

$$
k^3 - 3k - 2 = 0
$$

This cubic factors as:

$$
(k + 1)^2(k - 2) = 0
$$

So the only positive root is $ k = 2 $.

---

### Step 4: Evaluate the Minimum

Substitute $ k = 2 $ into the expression:

$$
f(2) = \frac{(2 \cdot 2 + 1)(8 \cdot 2^2 + 16 \cdot 2 + 16)}{2^2} = \frac{5 \cdot (32 + 32 + 16)}{4} = \frac{5 \cdot 80}{4} = \frac{400}{4} = 100
$$

---

### Step 5: Confirm Optimality

Since the function is symmetric in $ a $ and $ b $, and the expression is homogeneous of degree 0, the minimum occurs when $ a = b $, and the value is minimized at $ k = 2 $. Further, as $ k \to 0^+ $ or $ k \to \infty $, the function tends to infinity, confirming that the critical point at $ k = 2 $ is indeed the global minimum.

---

### Final Answer

$$
\boxed{100}
$$

Solution:
By AM-GM,
\[a + b \ge 2 \sqrt{ab},\]so $(a + b)^2 \ge 4ab.$

Also by AM-GM,
\[(a + 2c) + (b + 2c) \ge 2 \sqrt{(a + 2c)(b + 2c)},\]so $(a + b + 4c)^2 \ge 4(a + 2c)(b + 2c).$

Hence,
\begin{align*}
(a + b)^2 + (a + b + 4c)^2 &\ge 4ab + 4(a + 2c)(b + 2c) \\
&= 8ab + 8ac + 8bc + 16c^2 \\
&= 8(ab + ac + bc + 2c^2).
\end{align*}By AM-GM,
\begin{align*}
ab + ac + bc + 2c^2 &= \frac{ab}{2} + \frac{ab}{2} + ac + bc + 2c^2 \\
&\ge 5 \sqrt[5]{\frac{ab}{2} \cdot \frac{ab}{2} \cdot ac \cdot bc \cdot 2c^2} \\
&= 5 \sqrt[5]{\frac{a^3 b^3 c^4}{2}}.
\end{align*}Also by AM-GM,
\begin{align*}
a + b + c &= \frac{a}{2} + \frac{a}{2} + \frac{b}{2} + \frac{b}{2} + c \\
&\ge 5 \sqrt[5]{\frac{a}{2} \cdot \frac{a}{2} \cdot \frac{b}{2} \cdot \frac{b}{2} \cdot c} \\
&= 5 \sqrt[5]{\frac{a^2 b^2 c}{16}}.
\end{align*}Hence,
\begin{align*}
\frac{(a + b + c)[(a + b)^2 + (a + b + 4c)^2]}{abc} &\ge 8 \cdot \frac{5 \sqrt[5]{\frac{a^2 b^2 c}{16}} \cdot 5 \sqrt[5]{\frac{a^3 b^3 c^4}{2}}}{abc} \\
&= 100.
\end{align*}Equality occurs when $a = b = 2$ and $c = 1,$ so the minimum value is $\boxed{100}.$