<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] Here I think this will work . Main idea being splitting up of $5x^2$ as $4x^2 + x^2$
$$(x+1)(x^4-x^3+5x^2-4x+4)=0$$
$$(x+1) (x^4-x^3+4x^2+ x^2-4x+4)=0 $$
$$(x+1) (x^4+4x^2+ x^2-x^3-4x+4)=0 $$
$$(x+1)[ x^2(x^2+4)+ (1-x)( x^2 +4)]=0 $$
$$(x+1)[ (x^2+4)(x^2+1-x))=0 $$
$$(x+1)[ (x^2+4)(x^2-x+1))=0 $$
Hence proved 
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