Example 14 Express $\frac{6 x^{2}-22 x+18}{(x-1)(x-2)(x-3)}$ as partial fractions.
$$
\text { Solution: Let } \begin{aligned}
\frac{6 x^{2}-22 x+18}{(x-1)(x-2)(x-3)} \\
\quad=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3},
\end{aligned}
$$

i.e.,
$$
\begin{array}{l}
6 x^{2}-22 x+18 \\
=A(x-2)(x-3)+B(x-3)(x-1) \\
\quad+C(x-1)(x-2) .
\end{array}
$$

Taking $x=1$, we get $2=2 A$, i.e., $A=1$;
Taking $x=2$, we get $-2=-B$, i.e., $B=2$;
Taking $x=3$, we get $6=2 C$, i.e., $C=3$.
$$
\begin{array}{r}
\therefore \frac{6 x^{2}-22 x+18}{(x-1)(x-2)(x-3)} \\
\quad=\frac{1}{x-1}+\frac{2}{x-2}+\frac{3}{x-3}
\end{array}
$$