Suppose we are given the fact that if $g(x)$ is irreducible and $g(x)$ divides both $f(x)$ and $f'(x)$, then $(g(x))^2$ divides $f(x)$. How do we factor $x^6+x^4+3x^2+2x+2$ into a product of irreducible over the complex numbers?
Attempt: I have tried the rational roots and there weren't any. I differentiated and integrating $f$ but a fifth and seventh degree polynomial wasn't any easier to work with as far as I can tell. I tried brute force long division by dividing some arbitrary $x^2+Ax+B$, and it was a mess. Currently I'm out of ideas.