Factorise completely the expression $(x + 2)^2$ - $(2x + 1)^2$

Analyze the question: This is a difference of two squares, a² − b², with a = (x + 2) and b = (2x + 1).

Concept: a² − b² = (a − b)(a + b)

Solution:

a − b = (x + 2) − (2x + 1) = x + 2 − 2x − 1 = 1 − x

a + b = (x + 2) + (2x + 1) = 3x + 3 = 3(x + 1)

So: (x + 2)² − (2x + 1)² = (1 − x) · 3(x + 1) = 3(x + 1)(1 − x)

Check by expanding: (x + 2)² = x² + 4x + 4 (2x + 1)² = 4x² + 4x + 1 Difference = −3x² + 3 = 3(1 − x²) = 3(1 − x)(1 + x) ✓

Why the other options are wrong:

• A: (3x + 2)(1 − x) expands to something with a −3x² term but a constant of 2, not 3. It fails because 3x + 3 was wrongly written as 3x + 2.
• B and C do not match the expansion −3x² + 3.

Answer: D

Exam tip: When you see one squared bracket minus another squared bracket, think "difference of two squares" right away. Always simplify each bracket (a − b and a + b) carefully, and take out common factors like the 3 in 3x + 3 to "factorise completely."