Chapter 6 - Section 6.3 - Simplifying Complex Fractions - Exercise Set - Page 361: 42

$\dfrac{1-4a}{4a+1}$

The given expression, $ \dfrac{a^{-1}-4}{4+a^{-1}} ,$ simplifies to \begin{array}{l}\require{cancel} \dfrac{\dfrac{1}{a^1}-4}{4+\dfrac{1}{a^1}} \\\\= \dfrac{\dfrac{1-4a}{a}}{\dfrac{4a+1}{a}} \\\\= \dfrac{\dfrac{1-4a}{\cancel{a}}}{\dfrac{4a+1}{\cancel{a}}} \\\\= \dfrac{1-4a}{4a+1} .\end{array}