<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] The answers actually
$$
(x^2 + 4)(x^3 + 1)\\
x^5 + 4x^3 + x^2 + 4\\
(x^5 + 4x^3)(x^2 + 4)\\
x^3(x^2 + 4) 1(x^2 + 4)\\
(x^2 + 4)(x^3 + 1)
$$
(Sorry, too lazy to actually change them to exponents)
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