#### Answer

$x=\left\{ -\dfrac{64}{3},4 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
(3x^2+52x)^{1/4}=4
,$ raise both sides to the exponent equal to $
4
.$ Then express the resulting equation in the form $ax^2+bx+c=0,$ and use the concepts of factoring quadratic equations to solve the resulting equation. Finally, do checking if the solution satisfies the original equation.
$\bf{\text{Solution Details:}}$
Raising both sides to the fourth power, the given equation becomes
\begin{array}{l}\require{cancel}
3x^2+52x=256
\\\\
3x^2+52x-256=0
.\end{array}
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
3(-256)=-768
$ and the value of $b$ is $
52
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-12,64
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
3x^2-12x+64x-256=0
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(3x^2-12x)+(64x-256)=0
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3x(x-4)+64(x-4)=0
.\end{array}
Factoring the $GCF=
(x-4)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(x-4)(3x+64)=0
.\end{array}
Equating each factor to zero (Zero Product Property), then
\begin{array}{l}\require{cancel}
x-4=0
\\\\\text{OR}\\\\
3x+64=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-4=0
\\\\
x=4
\\\\\text{OR}\\\\
3x+64=0
\\\\
3x=-64
\\\\
x=-\dfrac{64}{3}
.\end{array}
Upon checking, $
x=\left\{ -\dfrac{64}{3},4 \right\}
$ satisfy the original equation.