10.2. Find all values of the parameter $a$ for which the equation $x^{2}+2 x+2|x+1|=a$ has exactly two roots.
Answer: $a>-1$. Solution. Write the equation in the form $(x+1)^{2}+2|x+1|=a+1$. Let $t=|x+1|, t \geq 0$. Then we get the quadratic equation $t^{2}+2 t=a+1 \Leftrightarrow(t+1)^{2}=a+2$. Considering that $t \geq 0$, the inequality $a+2 \geq 1$ must be satisfied, i.e., $a \geq-1$. Under this condition, $t$ is uniquely determined (considering $t \geq 0$): $t=\sqrt{a+2}-1$. When $a=-1$, the original equation has a unique solution $x=-1$, and when $a>-1$, there are two solutions: $x_{1,2}= \pm(\sqrt{a+2}-1)-1$.