Answer
$x=\left\{ -\dfrac{1}{5},1 \right\}$
Work Step by Step
Using $x^{-a}=\dfrac{1}{x^a}$, the given equation, $
p^{-2}+4p^{-1}-5=0
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{p^2}+\dfrac{4}{p}-5=0
.\end{array}
Multiplying both sides by the $LCD=
p^2
,$ then the solution to the given equation is
\begin{array}{l}\require{cancel}
1(1)+p(4)-p^2(5)=0
\\\\
1+4p-5p^2=0
\\\\
-5p^2+4p+1=0
\\\\
5p^2-4p-1=0
\\\\
(5p+1)(p-1)=0
\\\\
x=\left\{ -\dfrac{1}{5},1 \right\}
.\end{array}
