To solve the expression [tex]\((2x + 2y)^2\)[/tex], we need to expand it step by step:
1. Write the expression as a product of two binomials:
[tex]\[
(2x + 2y)^2 = (2x + 2y)(2x + 2y)
\][/tex]
2. Now expand the product using the distributive property (also known as the FOIL method for binomials). Multiply each term in the first binomial by each term in the second binomial:
[tex]\[
(2x + 2y)(2x + 2y) = 2x \cdot 2x + 2x \cdot 2y + 2y \cdot 2x + 2y \cdot 2y
\][/tex]
3. Calculate each term separately:
[tex]\[
2x \cdot 2x = 4x^2
\][/tex]
[tex]\[
2x \cdot 2y = 4xy
\][/tex]
[tex]\[
2y \cdot 2x = 4yx = 4xy
\][/tex]
[tex]\[
2y \cdot 2y = 4y^2
\][/tex]
4. Add all the terms together:
[tex]\[
4x^2 + 4xy + 4xy + 4y^2
\][/tex]
5. Combine like terms:
[tex]\[
4x^2 + 8xy + 4y^2
\][/tex]
Thus, the expanded expression is:
[tex]\[
4x^2 + 8xy + 4y^2
\][/tex]
Looking at the choices provided, we see that our expanded expression corresponds to Choice (B):
[tex]\[
\boxed{4x^2 + 4xy + 4y^2 = 4x^2 + 8xy + 4y^2}
\][/tex]
