# 5. Problem: Solve the equation

$$
\left(x^{3}+x^{2}+x+1\right)\left(x^{11}+x^{10}+\ldots+x+1\right)=\left(x^{7}+x^{6}+\ldots+x+1\right)^{2}
$$
Solution: Multiply both sides of the equation by $(x-1)^{2}$, then the equation will take the form:

$\left(x^{4}-1\right)\left(x^{12}-1\right)=\left(x^{8}-1\right)^{2}$, expand the brackets:

$x^{16}-x^{12}-x^{4}+1=x^{16}-2 x^{8}+1$

$x^{12}-2 x^{8}+x^{4}=0$

$x^{4}\left(x^{8}-2 x^{4}+1\right)=0$

$x^{4}\left(x^{4}-1\right)^{2}=0$

$x=0,1,-1$.

The root $x=1$ could have arisen due to multiplication by $(x-1)^{2}$, and by checking, we confirm that $x=1$ is not a root of the original equation.

Answer: $-1,0$.