Which is the correct factorised form of x² − 2xy + y² − z²?

Correct answer: (x − y − z)(x − y + z)

The correct answer is A: (x − y − z)(x − y + z). First inspect the first three terms: x² − 2xy + y² = (x − y)², using p² − 2pq + q² = (p − q)². The whole expression becomes (x − y)² − z². Now apply the difference-of-squares identity r² − s² = (r − s)(r + s), with r = x − y and s = z. This gives [(x − y) − z][(x − y) + z] = (x − y − z)(x − y + z). Option B uses (x + y)² and would produce +2xy, not −2xy. Option C is not fully factorised and also has z instead of z². Option D leaves the expression partly expanded rather than factorised into the required product. A useful check is to multiply the two factors: the z terms cancel and the original four-term expression returns.