#### Answer

$x=\left\{ -4,2 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given eqution, $
x^2+2x=8
,$ express the equation in the form $ax^2+bx+c=0.$ Then express the equation in factored form. Next step is to equate each factor to zero (Zero Product Property). Finally, solve each equation.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x^2+2x-8=0
.\end{array}
In the trinomial expression above, the value of $c$ is $
-8
$ and the value of $b$ is $
2
.$
The possible pairs of integers whose product is $c$ are
\begin{array}{l}\require{cancel}
\{ 1,-8 \}, \{ 2,-4 \},
\\
\{ -1,8 \}, \{ -2,4 \}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-2,4
\}.$
Hence, the factored form of the expression above is
\begin{array}{l}\require{cancel}
(x-2)(x+4)=0
.\end{array}
Equating each factor to zero (Zero Product Property), the solutions to the equation above are
\begin{array}{l}\require{cancel}
x-2=0
\\\\\text{OR}\\\\
x+4=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-2=0
\\\\
x=2
\\\\\text{OR}\\\\
x+4=0
\\\\
x=-4
.\end{array}
Hence, $
x=\left\{ -4,2 \right\}
.$