Factorise $x^5+x+1$ 
Factorise $$x^5+x+1$$

I'm being taught that one method to factorise this expression which is $=x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1$
$=x^3(x^2+x+1)-x^2(x^2+x+1)+x^2+x+1$
=$(x^3-x^2+1)(x^2+x+1)$
My question:
Is there another method to factorise this as this solution it seems impossible to invent it?

Let $f (x)=x^5+x+1$
$$f'(x)=5x^4+1>0$$
$f $ is stricly increasing at $\mathbb R $, thus it has only one real root $\alpha \in (-1,0) $.
the factorisation will be of the form
$$(x-\alpha)(x^2+ax+b)(x^2+cx+d) $$
with $a^2-4b <0$ and $c^2-4d <0$.
