Factoring into Linear Factors

Question

How would one factor the following expression:

$(b - a)(c^2 - a^2) - (c-a)(b^2 - a^2)$

into the set of linear factors:

$(b - a)(c - a)(c - b)$

(This is not for homework but rather exam review. I ran into this issue when required to calculate a matrix's determinant in linear factor form).

Thank you!

Answer 1

$(b - a)(c^2 - a^2) - (c-a)(b^2 - a^2) = (b - a)(c+a)(c-a) - (c-a)(b+a)(b-a) = (b-a)(c-a)\left[(c+a) - (b+a)\right] = (b-a)(c-a)(c-b)$

Answer 2

Recall that $$x^2-y^2=(x-y)(x+y)$$ then \begin{array}((b - a)(c^2 - a^2) - (c-a)(b^2 - a^2)&=(b-a)(c-a)(c+a)-(c-a)(b-a)(b+a)\\&=(b-a)(c-a)[(c+a)-(b+a)]\\&=(b-a)(c-a)(c-b)\end{array}