Which is the correct factorisation of \(a^2 − b^2 − a + b\)?

Correct answer: (a − b)(a + b − 1)

The direct answer is A. Start with the difference of squares: \(a^2−b^2=(a−b)(a+b)\). The remaining terms are \(−a+b\), which equal \(−(a−b)\), not \(−(a+b)\). So the whole expression becomes \((a−b)(a+b)−(a−b)\). Taking \((a−b)\) common gives \((a−b)[a+b−1]=(a−b)(a+b−1)\). Option B expands to \(a^2−b^2−a−b\), so the sign of b is wrong. Option C expands to \(a^2−b^2+a−b\), so the signs of the remaining linear terms are wrong. Option D is a square and produces different cross terms and coefficients. A common mistake is to mishandle \(−a+b\); carefully factor it as \(−(a−b)\). Memory cue: first use \(a^2−b^2\), then look for the same bracket in the leftover linear terms.