= Solution
Suppose $m<n$. The <Rodrigues' formula> and $n$ integrations by parts give
$$
\begin{aligned}
\langle p_n,p_m\rangle
&=(-1)^n\int_0^\infty
p_m(x)\frac{d^n}{dx^n}
\left(x^{n+1/2}e^{-x}\right)dx\\
&=\int_0^\infty
p_m^{(n)}(x)x^{n+1/2}e^{-x}\,dx=0,
\end{aligned}
$$
because $p_m^{(n)}=0$. Every boundary term vanishes: the exponential controls infinity, while the remaining power of $x$ has positive exponent at zero. Symmetry of the <inner product> gives orthogonality whenever $m\ne n$.
For the norm, monicity gives $p_n^{(n)}=n!$, so the same calculation yields
$$
\boxed{
\langle p_n,p_n\rangle
=n!\int_0^\infty x^{n+1/2}e^{-x}\,dx
=n!\,\Gamma\!\left(n+\frac32\right)
}.
$$
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