Example 5. Factorize in the set of real numbers $R$:
$$
(x-1)(x-2)(x-3)(x-4)-24
$$
\begin{aligned} \text { Sol } & \text { Original expression }=[(x-1)(x-4)][(x-2) \\ & \cdot(x-3)]-24 \\ = & \left(x^{2}-5 x+4\right)\left[\left(x^{2}-5 x+4\right)+2\right]-24 \\ = & \left(x^{2}-5 x+4\right)^{2}+2\left(x^{2}-5 x-4\right)-24 \\ = & {\left[\left(x^{2}-5 x+4\right)+6\right]\left[\left(x^{2}-5 x+4\right)-4\right] } \\ = & x\left(x^{2}-5 x+10\right)(x-5) .\end{aligned}