<s>[INST] 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?
 [/INST] With algebraic identities, this is actually rather natural:
Remember if we had all powers of $x$ down to $x^0=1$, we could easily factor:
$$x^5+x^4+x^3+x^2+x+1=\frac{x^6-1}{x-1}=\frac{(x^3-1)(x^3+1)}{x-1}=(x^2+x+1)(x^3+1),$$ so we can write:
\begin{align}x^5+x+1&=(x^2+x+1)(x^3+1)-(x^4+x^3+x^2)\\
&=(x^2+x+1)(x^3+1)-x^2(x^2+x+1)\\
&=(x^2+x+1)(x^3+1-x^2).
\end{align}
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