Is the set $S=\{ p(x) e^{-\alpha \|x\|^2} : p(x) \in \mathcal{P}, \alpha >0 \}$ dense in $L^2$ where $\mathcal{P}$ is a set of polynomials.

Question

Let $\mathcal{P}$ be the set of all realvalued polynomials on $\mathbb{R}^2$ and define $$S=\{ p(x) e^{-\alpha \|x\|^2}, x\in \mathbb{R}^2 : p \in \mathcal{P}, \alpha >0 \}.$$

I have two questions, one minor and one major:

1. (Minor) Is $S$ is a dense subset of $L^2(\mathbb{R}^2)$?
2. (Major) What are some minimal assumptions on $\mathcal{P}$ that guarantee that $S$ is dense? For example, is it enough to consider only all homogeneous polynomials plus a constant.

I think the answer to the first question follows from here where this result was shown for a one-dimensional case.