Example 3 Solve the equation:
$$
|2 x-3|=2 \sqrt{(x-5)(x+2)}+1 .
$$
Solution: According to the problem,
$$
|2 x-3|-2 \sqrt{(x-5)(x+2)}=1 \text {, }
$$

which is $|2 x-3|-\sqrt{4 x^{2}-12 x-40}=1$,
$$
\sqrt{(2 x-3)^{2}}-\sqrt{(2 x-3)^{2}-49}=1 \text {. }
$$

Let $\sqrt{(2 x-3)^{2}}=m, \sqrt{(2 x-3)^{2}-49}=$ $n$, then
$$
\left\{\begin{array}{l}
m-n=1, \\
m^{2}-n^{2}=49 .
\end{array}\right.
$$

By $(2) \div(1)$ we get
$$
m+n=49 \text {. }
$$
Solving (1) and (3) together, we get $\left\{\begin{array}{l}m=25 \text {, } \\ n=24 \text {. }\end{array}\right.$

Thus, $\sqrt{(2 x-3)^{2}}=25$.
Solving for $x$ gives $x_{1}=-11, x_{2}=4$.
Upon verification, $x_{1}=-11, x_{2}=4$ are both roots of the original equation.