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).

$$ab^3-a^3b+bc^3-b^3c+ac^3-a^3c$$ $$=ab(a+b)(b-a)+bc (b+c)(c-b)+ac (a+c)(a-c)$$ $$=ab( (a+b+c)-c)(b-a)+bc ((a+b+c)-a)(c-b)+ac  ((a+b+c)-b)(a-c)$$ $$=(a+b+c) [ab (b-a)+bc (c-b)+ac (a-c)]-abc (b-a+c-b+a-c)$$ $$=(a+b+c) [ab (b-a)+bc (c-b)+ac (a-c)]+0$$$$=(a+b+c)[(a-b)(b-c)(c-a)]$$
