Problem 11.1. Solve the equation

$$
\log _{a}\left(a^{2\left(x^{2}+x\right)}+a^{2}\right)=x^{2}+x+\log _{a}\left(a^{2}+1\right)
$$

where $a$ is a real number.

Emil Kolev
11.1. It is clear that $a>0, a \neq 1$. We have

$$
a^{x^{2}+x}\left(a^{2}+1\right)=a^{2\left(x^{2}+x\right)}+a^{2}
$$

Setting $u=a^{x^{2}+x}$ gives the equation

$$
u^{2}-\left(a^{2}+1\right) u+a^{2}=0
$$

with roots 1 и $a^{2}$. Then $x^{2}+x=0$ and $x^{2}+x-2=0$, respectively. Thus, for any $a>0, a \neq 1$, the equation has four roots $x=-2,-1,0,1$.