How to factor intricate polynomial $ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $ I would like to know how to factor the following polynomial. 
$$ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $$
What is the method I should use to factor it? If anyone could help.. Thanks in advance.

Let
$$
f(a)=ab^3-a^3b+a^3c-ac^3+bc^3-b^3c.
$$
If $a=b$, then $f(a)=0$. Therefore $(a-b)\mid f(a)$. Analogously $(a-c)\mid f(a)$; so, $(a-b)(a-c)\mid f(a)$. But we can take it directly:
$$
ab^3-a^3b+a^3c-ac^3+bc^3-b^3c = ab(b^2-a^2) + c^3(b-a) - c(b^3-a^3) =\\= 
ab(b-a)(b+a)+c^3(b-a)-c(b-a)(b^2+ba+a^2) =\\= (b-a)[ab(b+a)+c^3-cb^2-cab-ca^2]
$$
$$
ab(b+a)+c^3-cb^2-cab-ca^2=ab(a+b) + c(c^2-a^2) - cb(a+b) =\\= (a-c)b(a+b) - c(a-c)(a+c)=(a-c)[b(a+b)-c(a+c)]
$$
And then
$$
b(a+b)-c(a+c) = ba+b^2 - ca - c^2 = a(b-c) + (b^2-c^2) =\\= a(b-c)+(b+c)(b-c)=(b-c)(a+b+c).
$$
So,
$$
ab^3-a^3b+a^3c-ac^3+bc^3-b^3c=(b-a)(a-c)(b-c)(a+b+c)
$$
