#### Answer

$\color{blue}{\left\{-\dfrac{64}{3}, 4\right\}}$

#### Work Step by Step

Raise both sides to the 4th power to obtain:
$((3x^2+52x)^{1/4})^4=4^4
\\3x^2+52x=256
\\3x^2+52x-256=0$
Factor the trinomial to obtain:
$(3x+64)(x-4)=0$
Use the Zero-Factor Property by equating each factor to zero, then solve each equation to obtain:
\begin{array}{ccc}
&3x+64=0 &\text{or} &x-4=0
\\&3x=-64 &\text{or} &x=4
\\&x=-\frac{64}{3} &\text{or} &x=4
\end{array}
Upon checking, both proposed solutions satisfy the original equation.
Thus, the solution set is $\color{blue}{\left\{-\dfrac{64}{3}, 4\right\}}$.