To factorize the given expression, we can use the distributive property of multiplication over addition. This means that we need to multiply each term in the first set of parentheses by each term in the second set of parentheses, and then simplify:
(2x + 2y)(x - y) + (2x - 2y)(x + y)
= 2x(x - y) + 2y(x - y) + 2x(x + y) - 2y(x + y) (using distributive property)
= 2x^2 - 2xy + 2xy + 2y^2 + 2x^2 + 2xy - 2xy - 2y^2 (combining like terms)
= 4x^2 - 4y^2
Therefore, the answer is 4(x - y)(x + y).
Explanation: We can factorize the given expression by grouping the terms into two sets, each set containing two terms with a common factor. We can then factor out that common factor from each set, and simplify. In this case, we can factor out 2 from the first set of parentheses, and -2 from the second set of parentheses, giving us:
2(x + y)(x - y) - 2(x + y)(x - y)
We can then combine the two sets of parentheses, giving us:
(2(x + y) - 2(x - y))(x - y)
Simplifying the expression inside the parentheses gives us:
(2x + 2y - 2x + 2y)(x - y)
Which further simplifies to:
4y(x - y)
However, this is not the fully factored form, since we can factor out 4 from the expression to get:
4(x - y)(y)
So the answer is 4(x - y)(y), which is equivalent to option A.