To simplify the given expression [tex]\((2x - 2y)(2y + 8)\)[/tex], we need to use the distributive property (also known as the FOIL method for binomials). Here is a step-by-step solution:
1. Distribute each term in the first binomial to each term in the second binomial:
[tex]\[
(2x - 2y)(2y + 8) = 2x \cdot 2y + 2x \cdot 8 - 2y \cdot 2y - 2y \cdot 8
\][/tex]
2. Perform the multiplications:
[tex]\[
2x \cdot 2y = 4xy
\][/tex]
[tex]\[
2x \cdot 8 = 16x
\][/tex]
[tex]\[
-2y \cdot 2y = -4y^2
\][/tex]
[tex]\[
-2y \cdot 8 = -16y
\][/tex]
3. Combine all the terms:
[tex]\[
4xy + 16x - 4y^2 - 16y
\][/tex]
So, the expression [tex]\((2x - 2y)(2y + 8)\)[/tex] simplifies to:
[tex]\[
4xy + 16x - 4y^2 - 16y
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{4xy + 16x - 4y^2 - 16y}
\][/tex]
Given the choices, the correct option is:
C. [tex]\(4 x y + 16 x - 4 y^2 - 16 y\)[/tex]
