#### Answer

$x=\left\{ -19,9 \right\}$

#### Work Step by Step

Using the properties of equality, the given equation, $
|x+5|-2=12
,$ is equivalent to
\begin{array}{l}\require{cancel}
|x+5|=12+2
\\\\
|x+5|=14
.\end{array}
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the equation above is equivalent to
\begin{array}{l}\require{cancel}
x+5=14 \text{ OR } x+5=-14
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x+5=14
\\\\
x=14-5
\\\\
x=9
\\\\\text{ OR }
\\\\
x+5=-14
\\\\
x=-14-5
\\\\
x=-19
.\end{array}
Hence, the solutions are $
x=\left\{ -19,9 \right\}
.$