If $G^{(k)}(y)$ is the $k^{th}$ derivative of the Gaussian function, how can we guarantee $|G^{(k)}| \leq C_k e^{-|y|}$ for some coefficients $C_k$?

Question

I came across the inequality in the question in Fourier Analysis and It's Applications by Folland, specifically in the section on Convolutions. If the Gaussian function is defined as

$$G(y) = \frac{1}{\sqrt{\pi}} e^{-y^2}\textit{,}$$

then

$$G^{(k)}(y) = P_k(y) e^{-y^2}\text{,}$$

for some polynomial $P_k(y)$. I'm not sure what kind of estimates are used to form the inequality $|G^{(k)}(y)| \leq C_k e^{-|y|}$. It's not a terribly hard thing to believe, since $e^{y^2}$ grows faster than any polynomial of finite degree, but I would like to understand the technical explanation for this.