= Solution
Use the <inner product> $\langle f,g\rangle=\mathbb E[f(Z)g(Z)]$ for a <standard normal random variable> $Z$. The degree-zero <monic polynomial> is $p_0=1$. Write $p_1=x+a$ and $p_2=x^2+bx+c$. The conditions
$$
0=\langle p_1,1\rangle=a,\qquad
0=\langle p_2,1\rangle=1+c,\qquad
0=\langle p_2,x\rangle=b
$$
use $\mathbb EZ=\mathbb EZ^3=0$ and $\mathbb EZ^2=1$. Hence these first three <Probabilists' Hermite polynomials> are
$$
\boxed{p_0(x)=1,\qquad p_1(x)=x,\qquad p_2(x)=x^2-1.}
$$
The <Gaussian fourth moment> is $\mathbb EZ^4=3$, so their squared <norms> are
$$
\boxed{h_0=1,\qquad h_1=1,\qquad
h_2=\mathbb E(Z^2-1)^2=3-2+1=2,}
\qquad
h_\ell=\int_{\mathbb R}p_\ell(x)^2\frac{e^{-x^2/2}}{\sqrt{2\pi}}\,dx.
$$
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