<s>[INST] $bc(b-c) + ca(c-a) + ab(a-b) = -(b-c)(c-a)(a-b)$ My book has used the equality $$bc(b-c) + ca(c-a) + ab(a-b) = -(b-c)(c-a)(a-b)$$ But its proof is not given.
When I open the LHS, I get $b^2c-bc^2+c^2a-ca^2+a^2b-ab^2$. I do not know how to proceed next.
 [/INST] $$bc(b-c)+ca(c-a)+ab(a-b)=b^2c-a^2c-bc^2+ac^2+ab(a-b)=$$
$$=c^2(a-b)-c(a-b)(a+b)+ab(a-b)=(a-b)(c^2-ac-bc+ab)=$$
$$=(a-b)(c-a)(c-b).$$
</s>