Example 4. Factorize
\[
\begin{aligned}
1+2 x+3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+5 x^{6}+4 x^{7} \\
+3 x^{8}+2 x^{9}+x^{10} .
\end{aligned}
\]
Solve for the non-1 roots of the polynomial $x^{5}=1$. Then its value is 0, similar to Example 2, the polynomial must be the same as the polynomial $x^{5}$
$$
\begin{array}{l}
+x^{4}+x^{3}+x^{2}+x+1 \text {, so } \\
\quad \text { the original expression }=\left(x^{2}+x^{4}+x^{3}+x^{2}+x+1\right)^{2} \\
=\left[x^{3}\left(x^{2}+x+1\right)+\left(x^{2}+x+1\right)\right]^{2} \\
=\left(x^{3}+1\right)^{2}\left(x^{2}+x+1\right)^{2} \\
=(x+1)^{2}\left(1-x+x^{2}\right)^{2}\left(1+x+x^{2}\right)^{2} .
\end{array}
$$