<s>[INST] Factoring the quintic polynomial $x^5+4x^3+x^2+4=0$ I am trying to factor
$$x^5+4x^3+x^2+4=0$$
I've used Ruffini's rule to get
$$(x+1)(x^4-x^3+5x^2-4x+4)=0$$
But I don't know what to do next.
The solution is $(x+1) (x^2+4) (x^2-x+1) = 0$. I've tried using the completing square method but with no result. Could you give me hints?
 [/INST] I would start by factoring $x^3$ out from the first two terms and noticing the pattern in the result.
$$
\begin{split}
x^5+4x^3 + x^2 + 4 &= x^3 \left(x^2+4\right) + x^2+4 \\
                   &= \left(x^2+4\right)\left(x^3+1\right) \\
                   &= \left(x^2+4\right)(x+1)\left(x^2-x+1\right), \\
\end{split}
$$
where the last step is the standard factoring of the sum of two cubes.
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