 (2a + 2b)³ + (a - b)³ = (3a + b) (Aa² + Bab + Cb²), then what is the value of (A + B - C)?

To find \( A + B - C \), let's expand both sides of the given equation and compare terms.

The equation is:
\[
(2a + 2b)^3 + (a - b)^3 = (3a + b)(A a^2 + B a b + C b^2)
\]

### Step 1: Expand the Left Side
First, let's expand each cube on the left-hand side separately.

#### Expanding \((2a + 2b)^3\):
\[
(2a + 2b)^3 = 8(a + b)^3 = 8(a^3 + 3a^2b + 3ab^2 + b^3)
\]
\[
= 8a^3 + 24a^2b + 24ab^2 + 8b^3
\]

#### Expanding \((a - b)^3\):
\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\]

Now, adding these two expansions together:
\[
(2a + 2b)^3 + (a - b)^3 = (8a^3 + 24a^2b + 24ab^2 + 8b^3) + (a^3 - 3a^2b + 3ab^2 - b^3)
\]
\[
= 9a^3 + 21a^2b + 27ab^2 + 7b^3
\]

### Step 2: Expand the Right Side
The right side is:
\[
(3a + b)(A a^2 + B a b + C b^2)
\]

Expanding this product:
\[
= 3a \cdot (A a^2 + B a b + C b^2) + b \cdot (A a^2 + B a b + C b^2)
\]
\[
= 3A a^3 + 3B a^2b + 3C ab^2 + A a^2b + B ab^2 + C b^3
\]
\[
= (3A) a^3 + (3B + A) a^2b + (3C + B) ab^2 + C b^3
\]

### Step 3: Equate the Terms
Now we match the coefficients of the terms on both sides of the equation:
- Coefficient of \(a^3\): \(3A = 9 \Rightarrow A = 3\)
- Coefficient of \(a^2b\): \(3B + A = 21 \Rightarrow 3B + 3 = 21 \Rightarrow 3B = 18 \Rightarrow B = 6\)
- Coefficient of \(ab^2\): \(3C + B = 27 \Rightarrow 3C + 6 = 27 \Rightarrow 3C = 21 \Rightarrow C = 7\)

### Step 4: Calculate \(A + B - C\)
Now that we have \(A = 3\), \(B = 6\), and \(C = 7\):
\[
A + B - C = 3 + 6 - 7 = 2
\]

### Final Answer:
\[
\boxed{2}
\]

Would you like further details or have any questions?

---

Here are 5 related questions to expand on this topic:
1. How can we verify our solution by substituting values of \(a\) and \(b\) into the equation?
2. What methods are useful in expanding binomials with large exponents?
3. Why is comparing coefficients a valid approach in polynomial equations?
4. How can we use similar problems to practice expanding and simplifying expressions?
5. What other techniques are there for simplifying complex expressions?

**Tip:** For problems with high-degree polynomials, practice organizing terms carefully to avoid errors in combining like terms.

(2a + 2b)³ + (a - b)³ = (3a + b) (Aa² + Bab + Cb²), then what is the value of (A + B - C)?

Explore a step-by-step solution to find the value of (A + B - C) in the polynomial expansion problem: (2a + 2b)^3 + (a - b)^3 = (3a + b)(Aa^2 + Bab + Cb^2). This advanced algebra problem uses polynomial expansion, binomial theorem, and coefficient comparison, ideal for Grades 11-12 students.

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