Show that $M:=\{e^{-\alpha \mid x \mid}\varphi : \varphi \in C_c^\infty (\Bbb R^n)\}$ is dense in $\mathrm L ^2 (\Bbb R^n, \lambda ^n)$

Question

Let $\alpha \gt 0$ and $n \in \Bbb N$. I have to show that the set $$M:=\{e^{-\alpha \mid x \mid}\varphi : \varphi \in C_c^\infty (\Bbb R^n)\}$$ is dense in $\mathrm L ^2 (\Bbb R^n, \lambda ^n)$

$\mathrm L^p (\Bbb R ^2, \lambda ^n ) := \mathcal L^p (\Bbb R ^2, \lambda ^n )/\mathcal N $ (with $ \mathcal N $ beeing the set of measurable functions f that vanish $\mu$ - a.e.) is thte factor vector space (quotient vector space).
$C_c^\infty (\Bbb R^n)$ is the space of all infinitely differetiable functions with compact support.

I'm really lost on how to even approach this.

Any tipps or ideas? Thanks in advance!