Answer
$x=-1$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given radical equation, $
\sqrt[3]{x^2+5x+1}=\sqrt[3]{x^2+4x}
,$ 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]{x^2+5x+1}\right)^3=\left(\sqrt[3]{x^2+4x}\right)^3
\\\\
x^2+5x+1=x^2+4x
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
(x^2-x^2)+(5x-4x)=-1
\\\\
x=-1
.\end{array}
Upon checking, $
x=-1
$ satisfies the original equation.