Answer
The simplified form of the rational expression $\left( {{x}^{2}}-16 \right)\div \frac{4x+16}{3{{x}^{2}}}$ is$\frac{3{{x}^{2}}\cdot \left( x-4 \right)}{4}$.
Work Step by Step
$\left( {{x}^{2}}-16 \right)\div \frac{4x+16}{3{{x}^{2}}}$
$\frac{A}{B}\div \frac{C}{D}=\frac{A}{B}\cdot \frac{D}{C}$
The reciprocal of $\frac{4x+16}{3{{x}^{2}}}$ is$\frac{3{{x}^{2}}}{4x+16}$.
So, multiply the reciprocal of the divisor,
$\begin{align}
& \left( {{x}^{2}}-16 \right)\div \frac{4x+16}{3{{x}^{2}}}=\left( {{x}^{2}}-16 \right)\cdot \frac{3{{x}^{2}}}{4x+16} \\
& =\frac{\left( {{x}^{2}}-16 \right)\left( 3{{x}^{2}} \right)}{4x+16}
\end{align}$
Factor the numerator:
$\begin{align}
& \left( {{x}^{2}}-16 \right)\left( 3{{x}^{2}} \right)=\left( {{x}^{2}}-{{4}^{2}} \right)\left( 3{{x}^{2}} \right) \\
& =\left( x+4 \right)\left( x-4 \right)\left( 3{{x}^{2}} \right)
\end{align}$
Factor the denominator as:
$4x+16=4\left( x+4 \right)$
So, the rational expression becomes,
$\left( {{x}^{2}}-16 \right)\div \frac{4x+16}{3{{x}^{2}}}=\frac{\left( x+4 \right)\left( x-4 \right)\left( 3{{x}^{2}} \right)}{4\left( x+4 \right)}$
Regroup and remove the factor equal to 1,
$\begin{align}
& \left( {{x}^{2}}-16 \right)\div \frac{4x+16}{3{{x}^{2}}}=\frac{\left( x+4 \right)\left( x-4 \right)\left( 3{{x}^{2}} \right)}{\left( x+4 \right)\left( 4 \right)} \\
& =\frac{\left( x+4 \right)}{\left( x+4 \right)}\cdot \frac{\left( x-4 \right)\left( 3{{x}^{2}} \right)}{4} \\
& =1\cdot \frac{\left( 3{{x}^{2}} \right)\left( x-4 \right)}{\left( 4 \right)} \\
& =\frac{3{{x}^{2}}\cdot \left( x-4 \right)}{4}
\end{align}$
