<s>[INST] Factorise the expression Factorise of the expression  $$ a^4+b^4-c^4-2a^2b^2+4abc^2~ \\(S:(a + b - c) (a + b + c) (a^2+ b^2 + c^2 - 2 a b))$$
I tried to solve this problem:
 $$ a^4+b^4-c^4-2a^2b^2+4abc^2\rightarrow\\\
(a^2-b^2)^2-c^4+4abc^2\rightarrow\\\
(a^2-b^2-c^2)(a^2-b^2+c^2)+4abc^2\\\
((a-b)(a+b)-c^2)((a-b)(a+b)+c^2)+4abc^2$$ but I stopped here. Can anyone help me?
 [/INST] As a quadratic in $c^2\,$, the reduced discriminant of $-(c^4-4ab \cdot c^2-a^4-b^4+2a^2b^2)$ is:
$$
\frac{1}{4}\Delta = 4a^2b^2 +a^4+b^4-2a^2b^2=(a^2+b^2)^2
$$
Therefore the roots are $c^2=2ab \pm (a^2+b^2) = \{(a+b)^2, -(a-b)^2\}\,$ giving the factorization:
$$
\begin{align}
-\left(c^2-(a+b)^2\right)\left(c^2+(a-b)^2\right) & = -(c-a-b)(c+a+b)(c^2+a^2-2ab+b^2) \\[3px]
 & = (a+b-c)(a+b+c)(a^2+b^2+c^2-2ab)
\end{align}
$$
</s>