Factorize $a^2 − b^2 − 4a + 4$

Analyze the question: We need to factorize a² − b² − 4a + 4. There are four terms, so we try grouping. Notice that a² − 4a + 4 is a perfect square.

Concept: Two key identities:

• Perfect square: x² − 2xy + y² = (x − y)²
• Difference of two squares: x² − y² = (x − y)(x + y)

Solution:

Step 1: Rearrange the terms so the ones in a are together: a² − 4a + 4 − b²

Step 2: Recognise the perfect square: a² − 4a + 4 = (a − 2)²

So the expression becomes: (a − 2)² − b²

Step 3: Apply the difference of two squares, with x = (a − 2) and y = b: (a − 2 − b)(a − 2 + b)

Check (quick expansion): (a − 2 + b)(a − 2 − b) = (a − 2)² − b² = a² − 4a + 4 − b² ✔

This matches option B, (a − 2 + b)(a − 2 − b).

Why the others are wrong:

• A: (a + b)(a − b) = a² − b², which is missing the −4a + 4.
• C: (a + 1)(a − 2 + b) does not expand to give our expression, since the first factor is wrong.
• D: (a + b)² = a² + 2ab + b², which is not the same.

Exam tip: When you see four terms with a squared term and a negative squared term (like −b²), look for three terms that form a perfect square (a² − 4a + 4) and then use the difference of two squares. Always expand quickly to verify.