*Gasp* this one is not easy.
Beginning with (a-b)(b-c)(c-a) is correct.
That will get us (ab-ac-b^2+bc)(c-a) which is ab^2-a^2b-ac^2-b^2c+bc^2.
And then we can see that, ab^2-a^2b-ac^2-b^2c+bc^2 times -(a+b+c) will get us ab^3-a^3b+bc^3-b^3c+ca^3-c^3a.
So, we combine this to the factor we already got: (a-b)(b-c)(c-a) and get -(a+b+c)(a-b)(b-c)(c-a), which is the same as -(a-b)(b-c)(c-a)(a+b+c)
