Polynomial-related manipulation My question is:

Factorize: $$x^{11} + x^{10} + x^9 + \cdots + x + 1$$

Any help to solve this question would be greatly appreciated.

$$
\begin{align}
& {}\quad (x^{11} + x^{10}) + (x^9 + x^8)+(x^7+x^6)+(x^5+x^4)+(x^3+x^2 )+( x + 1)\\[8pt]
& =x^{10}(x+1)+x^8(x+1)+x^6(x+1)+x^4(x+1)+x^2(x+1)+(x+1)\\[8pt]
& =(x+1)(x^{10}+x^8+x^6+x^4+x^2+1)\\[8pt]
& =(x+1)(x^8(x^2+1)+x^4(x^2+1)+x^2+1)\\[8pt]
& =(x+1)((x^2+1)(x^8+x^4+1))\\[8pt]
& =(x+1)(x^2+1)(x^4+1-x^2)(x^4+1+x^2)\\[8pt]
& =(x+1)(x^2+1)(x^4+1-x^2)(x^2+1-x)(x^2+1+x)
\end{align}
$$
