Example 2 Given that $a$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$. Then the value of $\frac{a^{3}-1}{a^{5}+a^{4}-a^{3}-a^{2}}$ is $\qquad$ .
$(1995$, National Junior High School Mathematics League)
Solution: According to the problem, we have $a^{2}+a=\frac{1}{4}$.
$$
\therefore \text { the original expression }=\frac{(a-1)\left(a^{2}+a+1\right)}{(a-1)\left(a^{2}+a\right)^{2}}=20 \text {. }
$$