Express the equation as:
[tex]\begin{gathered} \sqrt[3]{x+6}-5=-2 \\ \end{gathered}[/tex]Add 5 to both sides:
[tex]\begin{gathered} \sqrt[3]{x+6}-5+5=-2+5 \\ \sqrt[3]{x+6}=3 \end{gathered}[/tex]Raise both sides to the 3rd power:
[tex]\begin{gathered} (\sqrt[3]{x+6})^3=3^3 \\ x+6=27 \end{gathered}[/tex]Subtract 6 from both sides:
[tex]\begin{gathered} x+6-6=27-6 \\ x=21 \end{gathered}[/tex]