2015/030) Factor $x^4 – 2ax^2+x+a^2-a$

It is not easy to solve a quartic equation but we see that it is quadratic in a

= $a^2 – a(2x^2+ 1) + x^4 + x$

now we try to factor $x^4 + x$ such that sum is $2x^2 + 1$

we have $x^4+ x = x(x^3+1) = x(x+1)(x^2- x + 1) = (x^2 + x)(x^2-x + 1)$

so the 2 factors are $x^2 + x$ and $x^2-x + 1$ as we see that sum is $2x^2+ 1$

so the original expression factors to $(a-x^2-x)(a-x^2+x-1)$ or $(x^2+x-a)(x^2-x+1-a)$