Example 5 Factorization:
$$
(x-1)(x-2)(x-3)(x-4)-120 \text {. }
$$
(1991, Jilin Province Participates in the National Junior High School Mathematics League Preliminary Selection Competition)
Solution: Let $f(x)$
$$
=(x-1)(x-2)(x-3)(x-4)-120 \text {. }
$$

Notice that
$$
\begin{array}{l}
f(-1)=(-2) \times(-3) \times(-4) \times(-5)-120=0, \\
f(6)=5 \times 4 \times 3 \times 2-120=0,
\end{array}
$$

Thus, we know that $f(x)$ has factors $x+1$ and $x-6$.
Expanding and performing division, we get
$$
\begin{array}{l}
f(x)=x^{4}-10 x^{3}+35 x^{2}-50 x-96 \\
=(x+1)(x-6)\left(x^{2}-5 x+16\right) .
\end{array}
$$