Which one of the following is a factor of a^{2}-b^{2}-c^{2}+2bc+a+b-c?

Explanation

Group the terms as (a^2 - (b^2+c^2-2bc)) + (a+b-c). This simplifies to (a^2 - (b-c)^2) + (a+b-c). Using difference of squares, we get (a - (b-c))(a + (b-c)) + (a+b-c) = (a-b+c)(a+b-c) + 1(a+b-c). Factoring out (a+b-c), it becomes (a+b-c)(a-b+c+1). Thus, (a-b+c+1) is a factor.

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