Factoring $ab^3 - a^3 b + bc^3 - b^3 c + ca^3 - c^3 a$ 
Factor $$ab^3 - a^3 b + bc^3 - b^3 c + ca^3 - c^3 a$$

I used the factor theorem to get factors
$$f(a, b, c)=(a-b)(b-c)(a-c)\;g (a, b, c)$$
for some polynomial $g (a, b, c)$.
How can I continue using this method?
(sorry for the previously messed up question I'm new to this website and didn't fully understand the guidelines).

@JohnBentin's method is the slickest, but if you don't want to check one example to get the scaling factor, let $x:=b^2-a^2,\,y:=c^2-b^2$ so your sum is$$\begin{align}abx+bcy-ca(x+y)&=a(b-c)x+c(b-a)y\\&=a(b-c)(b-a)(b+a)+c(b-a)(c-b)(c+b)\\&=(b-a)(c-b)(c(c+b)-a(b+a))\\&=(b-a)(c-b)(c^2+bc-ab-a^2)\\&=(b-a)(c-b)(c-a)(a+b+c).\end{align}$$In fact, his method helps you guess some components of this calculation as you go, so it's not as taxing as it looks.
