Example 1. Factorize:
$$
\left(x^{2}-x+15\right)\left(x^{2}-x-5\right)+51
$$
\begin{aligned} \text { Sol } & \text { Let } y=\frac{x^{2}-x+15+x^{2}-x-5}{2} \\ = & x^{2}-x+5 . \text { Then } \\ \text { Original expression } & =(y+10)(y-10)+51=y^{2}-49 \\ & =(y-7)(y+7) \\ & =(x-2)(x+1)\left(x^{2}-x+12\right) .\end{aligned}