Example 9 Express $\frac{11 x^{2}-23 x}{(2 x-1)\left(x^{2}-9\right)}$ as partial fractions.
Solution: Let $N=\frac{11 x^{2}-23 x}{(2 x-1)(x+3)(x-3)}$
$$
\equiv \frac{A}{2 x-1}+\frac{B}{x+3}+\frac{C}{x-3} \text {. }
$$

Then $11 x^{2}-23 x$
$$
\begin{aligned}
\equiv & A(x+3)(x-3)+B(2 x-1)(x-3)+ \\
& C(2 x-1)(x+3) .
\end{aligned}
$$

Let $x=\frac{1}{2}$, substituting we get $A=1$;
Let $x=-3$, substituting we get $B=4$;
Let $x=3$, substituting we get $C=1$.
Therefore, $N=\frac{1}{2 x-1}+\frac{4}{x+3}+\frac{1}{x-3}$. (Can be verified by combining the fractions)