Example 2 Solve the system of inequalities: $\left\{\begin{array}{l}x^{2}>x+2, \\ 4 x^{2} \leqslant 4 x+15 .\end{array}\right.$
Solve the system of inequalities
$$
\left\{\begin{array}{l}
x^{2}>x+2 \\
4 x^{2} \leqslant 4 x+15
\end{array}\right.
$$

we get

Therefore,
$$
\begin{array}{c}
\left\{\begin{array}{l}
(x+1)(x-2)>0, \\
(2 x+3)(2 x-5) \leqslant 0,
\end{array}\right. \\
\left\{\begin{array}{l}
x<-1 \text{ or } x>2, \\
-\frac{3}{2} \leqslant x \leqslant \frac{5}{2} .
\end{array}\right.
\end{array}
$$

Thus, the solution set of the original system of inequalities is $\left[-\frac{3}{2},-1\right) \cup\left(2, \frac{5}{2}\right]$.