Question

which expression is equivalent to this polynomial expression?
$(2a - 3b)(2a^{2} + 6b^{2}) + (6a^{2}b + 12ab^{2} - 7b^{3})$
a. $4a^{3} + 12a^{2}b - 25b^{3}$
b. $4a^{3} - 12a^{2}b - 24ab^{2} - 11b^{3}$
c. $4a^{3} + 24ab^{2} - 25b^{3}$
d. $4a^{3} + 12a^{2}b + 24ab^{2} - 11b^{3}$

Explanation:

Step1: Expand the first product

Use distributive property (FOIL) on $(2a - 3b)(2a^2 + 6b^2)$:

$$\begin{align*} 2a(2a^2 + 6b^2) - 3b(2a^2 + 6b^2) &= 4a^3 + 12ab^2 - 6a^2b - 18b^3 \end{align*}$$

Step2: Add the second polynomial

Combine the expanded result with $(6a^2b + 12ab^2 - 7b^3)$:

$$\begin{align*} &4a^3 + 12ab^2 - 6a^2b - 18b^3 + 6a^2b + 12ab^2 - 7b^3 \end{align*}$$

Step3: Combine like terms

Group and simplify terms with the same variables:

$$\begin{align*} &4a^3 + (-6a^2b + 6a^2b) + (12ab^2 + 12ab^2) + (-18b^3 - 7b^3)\\ &= 4a^3 + 0a^2b + 24ab^2 - 25b^3 \end{align*}$$

Answer:

C. $4a^3 + 24ab^2 - 25b^3$