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?

standard trick from contests: $5 \equiv 2 \pmod 3.$ Therefore, if $\omega^3 = 1$ but $\omega \neq 1,$ we get 
$$ \omega^5 = \omega^2  $$
$$ \omega^5 + \omega + 1 = \omega^2 + \omega + 1 = 0  $$
Which means, various ways of saying this, $x^5 + x + 1$ must be divisible by
$$ (x - \omega)(x - \omega^2) = x^2 + x + 1. $$
This is what we call a "minimal polynomial" for $\omega$
The same idea would work for
$$ x^{509} + x^{73} + 1  $$
although the other factor would be worse
SEE
Prove that $n^5+n^4+1$ is composite for $n>1.$ 
Prime factor of $A=14^7+14^2+1$
