1. Given $A=\left\{x \mid x^{3}+3 x^{2}+2 x>0\right\}, B=\left\{x \mid x^{2}+a x+b \leqslant 0\right\}$, and $A \cap B=\{x \mid 0<x \leqslant 2\}$. Find the values of $a, b$.
1. Solve $A=\{x \mid-20\}, B=\left[x_{1}, x_{2}\right]$. From $A \cap B=(0,2]$ we know:
$$
x_{2}=2 \text {, and } -1 \leqslant x_{1} \leqslant 0
$$

From $A \cup B=(-2,+\infty)$ we know:
$$
-2<x_{1} \leqslant-1
$$

Therefore, $x_{1}=-1$.
So $a=-\left(x_{1}+x_{2}\right)=-1, b=x_{1} x_{2}=-2$.