Answer
$\left\{-6,-2\right\}$
Work Step by Step
Multiplying by the $LCD=
x
,$ the given equation $
\dfrac{-12}{x}=x+8
,$ is equivalent to
\begin{align*}\require{cancel}
x\left(\dfrac{-12}{x}\right)&=(x+8)x
\\\\
-12&=x^2+8x
\\
0&=x^2+8x+12
\\
x^2+8x+12&=0
.\end{align*}
Using the factoring of trinomials, the factored form of the equation above is
\begin{align*}
(x+6)(x+2)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
x+6=0 & x+2=0
\\
x=-6 & x=-2
.\end{array}
Checking the solutions in the given equation results to
\begin{array}{l|r}
\text{If }x=-6: & \text{If }x=-2:
\\\\
\dfrac{-12}{-6}\overset{?}=-6+8 &
\dfrac{-12}{-2}\overset{?}=-2+8
\\\\
2\overset{\checkmark}=2 &
6\overset{\checkmark}=6
.\end{array}
Since both solutions satisfy the given equation, then the solution set of the equation $
\dfrac{-12}{x}=x+8
$ is $\left\{-6,-2\right\}$.
