<s>[INST] 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
 [/INST] the resultat should be $$- \left( b-c \right)  \left( a-c \right)  \left( a-b \right)  \left( a
+c+b \right) 
$$
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