Okay, so I need to find the minimum value of this expression: [(a + b + c)( (a + b)^2 + (a + b + 4c)^2 ) ] divided by (abc), where a, b, c are positive real numbers. Hmm, this looks like an optimization problem with constraints. Since all variables are positive, maybe I can use techniques from calculus, like partial derivatives, but that might get complicated. Alternatively, maybe inequalities like AM-GM could help here. Let me think.

First, let me try to simplify the expression a bit. Let's denote S = a + b + c. Then, the numerator becomes S * [ (a + b)^2 + (a + b + 4c)^2 ]. Let's expand that part inside the brackets:

(a + b)^2 + (a + b + 4c)^2 = (a + b)^2 + (a + b)^2 + 8c(a + b) + 16c^2 = 2(a + b)^2 + 8c(a + b) + 16c^2.

So, the entire numerator is S * [2(a + b)^2 + 8c(a + b) + 16c^2]. Since S = a + b + c, maybe we can express everything in terms of a + b and c. Let me let x = a + b and y = c. Then, S = x + y. The numerator becomes:

(x + y) * [2x^2 + 8xy + 16y^2] = (x + y)(2x^2 + 8xy + 16y^2).

The denominator is abc. But with x = a + b and y = c, we still have abc in terms of a, b, c. Maybe this substitution isn't helpful. Alternatively, perhaps I can set variables in such a way that some variables are equal? For example, maybe a = b? Let me check if symmetry is possible here.

If I assume a = b, then x = 2a, and c = y. Then S = 2a + y. The numerator becomes (2a + y)(2*(2a)^2 + 8*(2a)*y + 16y^2) = (2a + y)(8a^2 + 16a y + 16y^2). Let's expand that:

= (2a + y)(8a^2 + 16a y + 16y^2)
= 2a*(8a^2) + 2a*(16a y) + 2a*(16y^2) + y*(8a^2) + y*(16a y) + y*(16y^2)
= 16a^3 + 32a^2 y + 32a y^2 + 8a^2 y + 16a y^2 + 16y^3
= 16a^3 + (32a^2 y + 8a^2 y) + (32a y^2 + 16a y^2) + 16y^3
= 16a^3 + 40a^2 y + 48a y^2 + 16y^3.

The denominator becomes a * a * y = a^2 y. So the entire expression is:

[16a^3 + 40a^2 y + 48a y^2 + 16y^3] / (a^2 y) = [16a^3 / (a^2 y)] + [40a^2 y / (a^2 y)] + [48a y^2 / (a^2 y)] + [16y^3 / (a^2 y)]
= 16a / y + 40 + 48 / a + 16y^2 / a^2.

Hmm, so if I set a = b, the problem reduces to minimizing 16a/y + 40 + 48/a + 16y²/a² with respect to a and y. Maybe I can combine these terms. Let me denote z = a/y, so a = z y. Then substituting back:

16(z y)/y + 40 + 48/(z y) + 16y²/(z² y²)
= 16z + 40 + 48/(z y) + 16/(z²).

But this seems to complicate things because we still have two variables, z and y. Maybe this substitution isn't helpful. Alternatively, perhaps another approach.

Wait, maybe I should use the method of Lagrange multipliers. Let's consider the function f(a, b, c) = [(a + b + c)( (a + b)^2 + (a + b + 4c)^2 ) ] / (abc). To find its minimum, we can set the partial derivatives with respect to a, b, and c equal to zero. However, this might get messy because of the high degree of the function. Let's see.

Alternatively, maybe using AM-GM inequality. Let's try to manipulate the expression to apply AM-GM. Let's note that the numerator has terms like (a + b + c) multiplied by a sum of squares. The denominator is abc, which suggests that we need to relate the numerator to abc in a way that allows us to apply AM-GM.

Let me try to write the expression as:

[(a + b + c)/(abc)] * [(a + b)^2 + (a + b + 4c)^2].

So, that's [ (1/(ab)) + (1/(ac)) + (1/(bc)) ] multiplied by [ (a + b)^2 + (a + b + 4c)^2 ].

Hmm, maybe not directly helpful. Alternatively, perhaps we can consider substituting variables. Let me set x = a + b, y = c. Then, the expression becomes:

(x + y) [x² + (x + 4y)² ] / (abc).

But abc is a bit tricky here. Since a + b = x, we can express a and b as x - c. Wait, no, a + b = x and c = y, so abc = a b c = a b y. But since a + b = x, maybe we can express ab in terms of x and some other variable. Let me recall that for two variables, ab ≤ (a + b)² / 4 by AM-GM. But here, we have ab in the denominator, so perhaps if we can bound ab from below.

Wait, but ab is in the denominator, so to minimize the overall expression, we need to maximize ab. Since ab ≤ (a + b)^2 / 4, the maximum ab is (x)^2 / 4. So, the minimum value of abc would be when ab is maximized, but that's if c is fixed. Hmm, but we need to relate all variables. Maybe this approach is not straightforward.

Alternatively, perhaps use substitution variables. Let me set t = c/(a + b). Then, c = t(a + b). Since c is positive, t is positive. Then, S = a + b + c = a + b + t(a + b) = (1 + t)(a + b). Let x = a + b. Then, S = (1 + t)x. The numerator becomes:

[(1 + t)x] [x² + (x + 4t x)^2 ] = (1 + t)x [x² + x²(1 + 4t)^2] = (1 + t)x^3 [1 + (1 + 4t)^2].

The denominator is abc = a b c. Since a + b = x, and c = t x, we can express ab in terms of x. Let me recall that for a + b = x, the product ab is maximized when a = b = x/2, so ab ≤ x² / 4. Therefore, abc = ab c ≤ (x² / 4)(t x) = (t x³)/4. Hence, the denominator is at most (t x³)/4, which implies that the entire expression is at least [ (1 + t)x^3 [1 + (1 + 4t)^2 ] ] / ( (t x³)/4 ) = 4(1 + t)[1 + (1 + 4t)^2 ] / t.

Therefore, the expression is greater than or equal to 4(1 + t)[1 + (1 + 4t)^2 ] / t. So, to find the minimum of the original expression, it's sufficient to minimize this lower bound over t > 0. Let's denote this lower bound as f(t):

f(t) = 4(1 + t)[1 + (1 + 4t)^2 ] / t.

Let me expand this:

First, compute (1 + 4t)^2 = 1 + 8t + 16t². Then, 1 + (1 + 4t)^2 = 2 + 8t + 16t². Therefore,

f(t) = 4(1 + t)(2 + 8t + 16t²) / t.

Let me expand (1 + t)(2 + 8t + 16t²):

= 1*(2 + 8t + 16t²) + t*(2 + 8t + 16t²)
= 2 + 8t + 16t² + 2t + 8t² + 16t³
= 2 + 10t + 24t² + 16t³.

So, f(t) = 4*(2 + 10t + 24t² + 16t³) / t.

Divide each term by t:

= 4*(2/t + 10 + 24t + 16t²).

Therefore, f(t) = 8/t + 40 + 96t + 64t².

Now, we need to find the minimum of f(t) = 64t² + 96t + 40 + 8/t for t > 0. Let's take the derivative of f(t) with respect to t:

f'(t) = 128t + 96 - 8/t².

Set derivative equal to zero:

128t + 96 - 8/t² = 0.

Multiply both sides by t² to eliminate denominator:

128t³ + 96t² - 8 = 0.

Divide equation by 8:

16t³ + 12t² - 1 = 0.

So, we need to solve 16t³ + 12t² - 1 = 0. Hmm, this cubic equation. Let's check for rational roots using Rational Root Theorem. Possible roots are ±1, ±1/2, ±1/4, etc. Let's test t = 1/2:

16*(1/2)^3 + 12*(1/2)^2 - 1 = 16*(1/8) + 12*(1/4) - 1 = 2 + 3 - 1 = 4 ≠ 0.

t = 1/4:

16*(1/4)^3 + 12*(1/4)^2 - 1 = 16*(1/64) + 12*(1/16) - 1 = (1/4) + (3/4) - 1 = 1 - 1 = 0. Oh! So t = 1/4 is a root.

Therefore, we can factor out (t - 1/4) from the cubic. Wait, but 16t³ + 12t² - 1. Let's perform polynomial division.

Divide 16t³ + 12t² - 1 by (t - 1/4). Wait, but t = 1/4 is a root, so factor is (t - 1/4). Let me write 16t³ + 12t² - 1 as (t - 1/4)(At² + Bt + C).

Multiply out (t - 1/4)(At² + Bt + C) = At³ + Bt² + Ct - (A/4)t² - (B/4)t - C/4.

Combine like terms:

= At³ + (B - A/4)t² + (C - B/4)t - C/4.

Set equal to 16t³ + 12t² - 1. Therefore, equate coefficients:

A = 16.

B - A/4 = 12 => B - 16/4 = B - 4 = 12 => B = 16.

C - B/4 = 0 => C - 16/4 = C - 4 = 0 => C = 4.

But the constant term is -C/4 = -4/4 = -1, which matches. Therefore, the cubic factors as (t - 1/4)(16t² + 16t + 4). Let's verify:

(t - 1/4)(16t² + 16t + 4) = t*(16t² + 16t + 4) - 1/4*(16t² + 16t + 4)
= 16t³ + 16t² + 4t - 4t² - 4t - 1
= 16t³ + (16t² - 4t²) + (4t - 4t) -1
= 16t³ + 12t² -1. Correct.

Therefore, 16t³ + 12t² -1 = (t - 1/4)(16t² + 16t + 4). The quadratic factor 16t² + 16t + 4 has discriminant 256 - 256 = 0. Wait, discriminant is (16)^2 - 4*16*4 = 256 - 256 = 0. So, it's a perfect square: 16t² + 16t + 4 = 4(4t² + 4t + 1) = 4(2t + 1)^2.

Therefore, the roots are t = 1/4, and t = -1/2 (double root). Since t > 0, the only critical point is t = 1/4.

Therefore, the minimum of f(t) occurs at t = 1/4. Let's compute f(1/4):

f(1/4) = 64*(1/4)^2 + 96*(1/4) + 40 + 8/(1/4)
= 64*(1/16) + 24 + 40 + 32
= 4 + 24 + 40 + 32
= 100.

Therefore, the minimum value of the original expression is 100. But wait, this is the lower bound we found earlier. However, we need to check if equality is achievable. That is, in the AM-GM step, we assumed that ab is maximized when a = b. So, when t = 1/4, which is c = t(a + b) = (1/4)(a + b). If we set a = b, then a + b = 2a, so c = (1/4)(2a) = a/2. Therefore, a = b, c = a/2. So, let's check if with these values, the original expression equals 100.

Let a = b = k, c = k/2. Then, compute the original expression:

[(a + b + c)( (a + b)^2 + (a + b + 4c)^2 )] / (abc)
= [(k + k + k/2)( (2k)^2 + (2k + 4*(k/2))^2 )] / (k * k * k/2)
= [(2k + k/2)(4k² + (2k + 2k)^2 )] / (k² * k/2)
= [(5k/2)(4k² + (4k)^2 )] / (k³/2)
= [(5k/2)(4k² + 16k²)] / (k³/2)
= [(5k/2)(20k²)] / (k³/2)
= (5k/2 * 20k²) / (k³/2)
= (100k³ / 2) / (k³ / 2)
= (50k³) / (k³ / 2)
= 50 * 2
= 100.

Yes! So the minimum value is indeed 100, achieved when a = b and c = a/2. Therefore, the answer is 100.

**Final Answer**
\boxed{100}
Therefore, the minimum value of the given expression is $\boxed{100}$.
</think>