Factorize completely; (2x + 2y)(x - y) + (2x - 2y)(x + y)

Analyze the question: We need to simplify and factorize (2x + 2y)(x − y) + (2x − 2y)(x + y) completely.

Concept: Take out common factors first, then combine like terms.

Solution:

Step 1: Factor out 2 from each bracket that has it.

• (2x + 2y) = 2(x + y)
• (2x − 2y) = 2(x − y)

Step 2: Substitute. 2(x + y)(x − y) + 2(x − y)(x + y)

Step 3: Both terms are identical, because multiplication is commutative: (x + y)(x − y) = (x − y)(x + y).

So we have 2(x + y)(x − y) + 2(x + y)(x − y) = 4(x + y)(x − y)

Check by expanding:

• (2x + 2y)(x − y) = 2x² − 2xy + 2xy − 2y² = 2x² − 2y²
• (2x − 2y)(x + y) = 2x² + 2xy − 2xy − 2y² = 2x² − 2y²
• Sum = 4x² − 4y² = 4(x² − y²) = 4(x − y)(x + y) ✔

Answer: Option A, 4(x − y)(x + y).

Tip: Always pull out common numbers first, then look for a difference of two squares, x² − y² = (x − y)(x + y). Checking by expanding takes only a few seconds and confirms your answer.