Q:

Factoring $(x+y+z)^3 xyz - (xz+xy+yz)^3$ and $(x-a)^3(b-c)^3+(x-b)^3(c-a)^3+(x-c)^3(a-b)^3$

I am trying to factor the expressions $(x+y+z)^3 xyz - (xz+xy+yz)^3$ and $(x-a)^3(b-c)^3+(x-b)^3(c-a)^3+(x-c)^3(a-b)^3$. 
I am rather stuck though.  Is there a general method for going about this?  I always find myself having to guess which is not so useful here.  I notice in the first one that if $x=y=z$ then the polynomial is zero.  I'm not sure how useful this is though in the case of three variables.  

A:

And
$$
(x - a)^3 (b - c)^3 + (x - b)^3 (c - a)^3 + (x - c)^3 (a - b)^3 = 3 (a - b) (a - c) (b - c) (a - x) (b - x) (c - x)
$$
This is easy to conclude because $a,b,c$ are roots

