Factorize; (2x + 3y)2 - (x - 4y)2

Analyze the question: This is a difference of two squares, where the expression has the form a² − b².

Concept: a² − b² = (a − b)(a + b)

Here, a = 2x + 3y and b = x − 4y.

Solution:

a − b = (2x + 3y) − (x − 4y) = 2x + 3y − x + 4y = x + 7y

a + b = (2x + 3y) + (x − 4y) = 3x − y

So: (2x + 3y)² − (x − 4y)² = (x + 7y)(3x − y)

Check by expanding:

• (2x + 3y)² = 4x² + 12xy + 9y²
• (x − 4y)² = x² − 8xy + 16y²
• Difference = 3x² + 20xy − 7y²

Expanding (3x − y)(x + 7y) = 3x² + 21xy − xy − 7y² = 3x² + 20xy − 7y² ✓

Answer: (3x − y)(x + 7y), which is option A.

Tip: When you see a squared bracket minus another squared bracket, use the difference of two squares straight away instead of expanding everything. Take care with the signs when subtracting the second bracket, since the minus sign changes both terms inside it (−(x − 4y) = −x + 4y). A quick expansion check at the end confirms your answer.