I want to prove that the following quadrature formula:
$$\int_{-\infty}^{+\infty} g(x)N(x|m, \sigma^2) dx \approx \frac{2}{3}g(m) +\frac{1}{6}g(m+\sigma\sqrt{3})+\frac{1}{6}g(m-\sigma\sqrt{3})$$ holds when g is polynomial less or equal to 5.
I managed to prove it for degree 0 with $g(x) = x^0 = 1$. We know that the integral of the normal gives 1.
However I am a bit lost starting from degree 1 where it is obviously easy to compute the RHS of the equality but I don't know how to deal with a product involving a normal. Using a software didn't help.