Factoring/Reducing a polynomial $x^4 -2x^3 + 2x^2 + x + 4$

Question

The problem asks to determine whether or not $x^4 -2x^3 + 2x^2 + x + 4$ is reducible in $\mathbb{Q}[x]$. I tried using the fact that if it is reducible (solution manual said it is) then it is reducible in $\mathbb{Z}[x]$. That didn't work for me. Eisenstein's criteria isn't applicable.

Answer 1

Since it is a monic polynomial, then the only possible rational roots are integer factors of its constant term. If none of those works, then it has no linear factor, so the only possible way to factor it is in the form $$(x^2+ax+b)(x^2+cx+d)$$ for some $a,b,c,d.$ Try expanding this product, equating the coefficients, and coming up with $a,b,c,d$ that fit the bill.