Example 3, Solve the equation: $\frac{4 x-5}{2 x-3}-\frac{15 x-23}{3 x-4}$
$$
=\frac{12 x-17}{6 x-10}-\frac{20 x-29}{4 x-5}
$$
Solution: By the division algorithm, we get
$$
\begin{aligned}
& \left(2+\frac{1}{2 x-3}\right)-\left(5-\frac{3}{3 x-4}\right) \\
= & \left(2+\frac{3}{6 x-10}\right)-\left(5-\frac{4}{4 x-5}\right) .
\end{aligned}
$$

Simplifying:
$$
\frac{1}{2 x-3}+\frac{3}{3 x-4}=\frac{3}{3 x-10}+\frac{4}{4 x-5} .
$$

Solving it yields $x=-\frac{10}{7}$.