Express $-a^3 b + a^3 c+ab^3-a c^3-b^3c+bc^3$ as a product of linear factors. 
Express
  $$-a^3 b + a^3 c+ab^3-a c^3-b^3c+bc^3$$
  as a product of linear factors.

I have tried rewriting the expression as:
$$ab^3-a^3b + a^3c-ac^3 +bc^3-b^3c$$
$$= ab(b^2-a^2)+ac(a^2-c^2)+bc(c^2-b^2)$$
$$= ab(b-a)(b+a) + ac(a-c)(a+c)+bc(c-b)(c+b)$$
Which is at least a product of linear factors. 
However I am now stuck as to how to proceed.
Also I would prefer the $(b^2-a^2)$ and the $(c^2-b^2)$ factors of my second line of working to be in the form $(a^2-b^2)$ and $(b^2-c^2)$, for 'neatness' , if possible

Seeing it as a polynomial of one variable might help.
$$\begin{align}&-a^3b+a^3c+ab^3-ac^3-b^3c+bc^3\\&=(c-b)a^3+(b^3-c^3)a+bc^3-b^3c\\&=(c-b)a^3+(b-c)(b^2+bc+c^2)a+bc(c-b)(c+b)\\&=(c-b)(a^3-(b^2+bc+c^2)a+bc(c+b))\\&=(c-b)((-a+c)b^2+(-ac+c^2)b+a^3-ac^2)\\&=(c-b)((c-a)b^2+c(c-a)b+a(a-c)(a+c))\\&=(c-b)(c-a)(b^2+bc-a(a+c))\\&=(c-b)(c-a)((b-a)(b+a)+c(b-a))\\&=(c-b)(c-a)(b-a)(a+b+c)\\&=(a-b)(b-c)(c-a)(a+b+c)\end{align}$$
