#### Answer

$r=-4$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given radical equation, $
\sqrt[3]{r^2+2r+8}=\sqrt[3]{r^2+3r+12}
,$ raise both sides of the equation to the third power. Then use properties of equality to isolate and solve the variable. Finally, do checking of the solution/s with the original equation.
$\bf{\text{Solution Details:}}$
Raising both sides of the equation to the third power results to
\begin{array}{l}\require{cancel}
\left(\sqrt[3]{r^2+2r+8}\right)^3=\left(\sqrt[3]{r^2+3r+12}\right)^3
\\\\
r^2+2r+8=r^2+3r+12
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
(r^2-r^2)+(2r-3r)=12-8
\\\\
-r=4
\\\\
r=\dfrac{4}{-1}
\\\\
r=-4
.\end{array}
Upon checking, $
r=-4
$ satisfies the original equation.