Example 2 Solve the inequality $\frac{8}{(x+1)^{3}}+\frac{10}{x+1}-x^{4}-5 x>0$.
Solving the high-order fractional inequality directly is not easy at first, but if we notice:
$$
\frac{8}{(x+1)^{3}}+\frac{10}{x+1}=\left(\frac{2}{x+1}\right)^{3}+5\left(\frac{2}{x+1}\right) .
$$

Then the original inequality can be transformed into: $\left(\frac{2}{x+1}\right)^{3}+5\left(\frac{2}{x+1}\right)>x^{3}+5 x$.
From this, we can construct the function $f(t)=t^{3}+5 t$, which is a monotonically increasing function on $\mathbf{R}$. The original inequality can be transformed into: $f\left(\frac{2}{x+1}\right)>f(x)$, so the original inequality is equivalent to: $\frac{2}{x+1}>x$. Solving this, we get the solution to the original inequality as $-1<x<1$ or $x<-2$.