Example 10. Express the fraction $\frac{7 x-8}{x^{2}+2 x-3}$ as partial fractions.
$$
\text { Let } \begin{aligned}
& \frac{7 x-8}{x^{2}+2 x-3}=\frac{A}{x+3}+\frac{B}{x-1} \\
= & \frac{(A+B) x+(3 B-A)}{x^{2}+2 x-3} .
\end{aligned}
$$

We have $\left\{\begin{array}{l}A+B=7, \\ 3 B-A=-8 .\end{array}\right.$

Thus,
$\left\{\begin{array}{l}A=\frac{29}{4}, \\ B=-\frac{1}{4} .\end{array}\right.$
Therefore, $\frac{7 x-8}{x^{2}+2 x-3}=\frac{29}{4(x+3)}-\frac{1}{4(x-1)}$.