To determine the system of equations that can be used to find the roots of the given equation [tex]\(4 x^2 = x^3 + 2 x\)[/tex], we need to rewrite the equation in a form that represents two separate functions in a system of equations.
Given the equation:
[tex]\[ 4x^2 = x^3 + 2x, \][/tex]
we can rewrite it as two distinct equations by setting them equal to a common variable [tex]\(y\)[/tex]. This means we will be looking to define the equation in a format such that:
[tex]\[ y = 4x^2 \][/tex]
and
[tex]\[ y = x^3 + 2x. \][/tex]
Therefore, the correct system of equations that represents this situation is:
[tex]\[
\left\{
\begin{array}{l}
y = 4x^2 \\
y = x^3 + 2x
\end{array}
\right.
\][/tex]
Among the provided options, this corresponds to:
[tex]\[
\left\{
\begin{array}{l}
y = 4x^2 \\
y = x^3 + 2x
\end{array}
\right.
\][/tex]
So the correct answer is:
[tex]\[
\left\{
\begin{array}{l}
y = 4x^2 \\
y = x^3 + 2x
\end{array}
\right.
\][/tex]
