Solution
= Solution
Let $x_1,\ldots,x_N$ be the distinct real roots of $\mathrm{He}_N$. The <Gauss-Hermite quadrature> rule is
$$
\int_{-\infty}^{\infty}f(x)e^{-x^2/2}\,dx
\approx\sum_{j=1}^Nw_jf(x_j),
$$
where $w_j$ is the weighted integral of the $j$th <Lagrange interpolation polynomial>. Equivalently,
$$
w_j=\frac{N!\sqrt{2\pi}}{N^2\mathrm{He}_{N-1}(x_j)^2}>0.
$$
The orthogonality of $\mathrm{He}_N$ makes the rule exact for every polynomial of degree at most
$$
\boxed{2N-1}.
$$