Name of distribution with density normal times polynomial (polynormal?)

Question

Is there a name for a probability distribution on $\mathbb{R}$ whose density is is given by $f(x)g(x)$, where $$f(x) = (2\pi \sigma)^{-1/2} \exp(-(x-\mu)^2/\sigma)$$ is the density of a normal distribution and $g(x)$ is polynomial in $x$? My specific use-case is where $$g(x) = (P + Qx^2).$$ My instinct is to call this "polynormal" but that might imply the law of $h(X)$, where $h$ is a polynomial and $X$ is normally distributed.