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?

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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.