Factoring polynomials again

Question

$bc(b+c)+ac(c+a)-ab(a+b)$

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Edit by @Andreas: This should be:

$bc(b+c)+ac(c-a)-ab(a+b)$

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Answer: $(a+b)(c-a)(b+c)$

I did: $b^{2}c+bc^{2}+ac^{2}+a^{2}c-a^{2}b-ab^{2}$ After this step I can't find a way to continue, could you give me a light on this solution?

Answer 1

You can proceed as follows (add and subtract the red b, then regroup):

$$ bc(b+c)+ac(c-a)-ab(a+b)\\ = bc(b+c)+ac(\color{red}b+c-(a+\color{red}b))-ab(a+b)\\ = (bc+ac)(b+c)- (ac+ab)(a+b)\\ = c(a+b)(b+c)- a(b+c)(a+b)\\ = (c-a)(a+b)(b+c) $$