Example 11 Given that $\alpha$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$. Find the value of $\frac{\alpha^{3}-1}{\alpha^{5}+\alpha^{4}-\alpha^{3}-\alpha^{2}}$.
(1995, National Junior High School Mathematics League)
Solution: From the given, we have $\alpha^{2}+\alpha=\frac{1}{4}$.
To form the required expression containing $\alpha^{2}+\alpha$, we have
$$
\frac{\alpha^{3}-1}{\alpha^{5}+\alpha^{4}-\alpha^{3}-\alpha^{2}}=\frac{(\alpha-1)\left(\alpha^{2}+\alpha+1\right)}{(\alpha-1)\left(\alpha^{2}+\alpha\right)^{2}}=20 .
$$