Factor $(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)$ to get $(b+c-a) (a+b-c) (a-b+c) (a+b+c)$ 
Is there an easy way to factor $(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)$ to get $(b+c-a) (a+b-c) (a-b+c) (a+b+c)$?

I have tried various methods such as completing the square, but I wasn't able to figure it out as it became very messy.

View the expression as a polynomial in $a$. Plug in $a=b+c$. If the polynomial vanishes $(b+c-a)$ is a factor of the polynomial. Repeat for other factors. As the degree of the polynomial is $4$, it can't have more than $4$ linear factors and you're done.
Note: This method is often used to factor multivariate polynomials and is only as good as your guess.
