<s>[INST] How to solve this algebraic manipulation problem? Express the following expression $$E=(x^3-y^3)(y^3-z^3)(z^3-x^3)$$ in terms of $a, b$ where $a,b \in \mathbb R$ and 
$$a=x^2y+y^2z+z^2x$$ $$b=xy^2+yz^2+zx^2$$
 [/INST] Start: $$ a-b = xy(x-y)+yz (y-z)+zx(z-x) $$
$$= xy(x-y)+yz (y-z)+zx(z-\color{red}y)+zx(\color{red}y-x) $$
$$= (xy-zx)(x-y)+(yz -zx)(y-z)$$
$$ =x(y-z)(x-y)+z(y-x)(y-z) $$
$$ = (x-y)(y-z)(x-z)$$
So $$ E = (a-b)\underbrace{(x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)}_{A}$$
Now you have to figer out $A$. (I bet it is $3ab$.)
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