Finding Greens Function using the delta function

Question

I have the linear differential operator $$Lu = u''-(1+x^2)u = f(x)$$ with BCs s.t. $u$ vanishes at positive and negative infinity. The hint I was given is to show that $u_0=e^{-x^2/2}$ satisfies $Lu_0=0$ and then find greens function of the form $wu_0$. I have tried this and end up with the $erf$ function as I need to integrate $e^{-t^2}$, this doesn't seem quite right so just wondering if anyone has any tips? Thanks!