Factorize: a² – b² + 2bc – c².

In the expression a² – b² + 2bc – c², observe that the last three terms form:- (b – c)².

⇒ a² – b² + 2bc – c² = a² – (b – c)²

We have the identity x² – y² = (x + y)(x – y)

⇒ a² – b² + 2bc – c² = [a + (b – c)][a – (b – c)]

∴ a² – b² + 2bc – c² = (a + b – c)(a – b + c)

Hence, the factors of a² – b² + 2bc – c² are (a + b – c) and (a – b + c).