Lining Up Critical Points

Let $f(x) = (x-a)^2(x+b)e^x$, where $a, b \in \mathbb{R}$.

1. When $a = 0$ and $b = -3$, find the intervals of monotonicity of $f(x)$. 2. When $a = 0$: if $x = 0$ is a local maximum point of $f(x)$, find the range of possible values of $b$. 3. Suppose $x = a$ is a local maximum point of $f(x)$, and let $x_1, x_2, x_3$ be the three critical points of $f(x)$. Do there exist a real $b$ and a real $x_4$ such that some ordering of $x_1, x_2, x_3, x_4$ forms an arithmetic sequence? If so, find all such $b$ and the corresponding $x_4$; if not, explain why.