Solve the equation $\left(1+x+x^{2}\right)\left(1+x+\ldots+x^{10}\right)=\left(1+x+\ldots+x^{6}\right)^{2}$.
Multiply both sides of the equation by $(1-x)^{2}:\left(1-x^{3}\right)\left(1-x^{11}\right)=\left(1-x^{7}\right)^{2}$.

After expanding the brackets, we get $-x^{3}-x^{11}=-2 x^{7}$, or $x^{3}\left(1-x^{4}\right)^{2}=0$. From this, $x=-1,0$ or 1. The root 1 could have arisen from multiplying by

$1-x$, checking confirms that this is indeed the case.

## Answer

$x=-1,0$.