7. Solve the inequality $(x+1)^{-x^{2}-2 x}>x+1$.

(20)
Solution. For $x+10$.

(6 points)

$1^{\circ}$ We sequentially obtain:

$$
\begin{array}{r}
-x^{2}-2 x0 \\
(x+1)^{2}>0
\end{array}
$$

which is satisfied for $x \neq -1$. In this case, the inequality is satisfied for $x \in (-1,0]$.

(6 points)

$2^{\circ}$ We obtain

$$
\begin{array}{r}
-x^{2}-2 x>1 \\
x^{2}+2 x+1<0 \\
(x+1)^{2}<0
\end{array}
$$

In this case, no $x$ satisfies the inequality.

(6 points)

The final solution is $x \in (-1,0]$.

(2 points)