Answer
The simplified form of the expression $\frac{2x\sqrt{{{x}^{2}}+5}-\frac{2{{x}^{3}}}{\sqrt{{{x}^{2}}+5}}}{{{x}^{2}}+5}$ is $\frac{10x}{\sqrt{{{\left( {{x}^{2}}+5 \right)}^{3}}}}$.
Work Step by Step
Consider the expression:
$\frac{2x\sqrt{{{x}^{2}}+5}-\frac{2{{x}^{3}}}{\sqrt{{{x}^{2}}+5}}}{{{x}^{2}}+5}$
Multiply the numerator as well as the denominator by $\sqrt{{{x}^{2}}+5}$
$\begin{align}
& \frac{2x\sqrt{{{x}^{2}}+5}-\frac{2{{x}^{3}}}{\sqrt{{{x}^{2}}+5}}}{{{x}^{2}}+5}=\frac{\left( 2x\sqrt{{{x}^{2}}+5}-\frac{2{{x}^{3}}}{\sqrt{{{x}^{2}}+5}} \right)\sqrt{{{x}^{2}}+5}}{{{x}^{2}}+5\cdot \sqrt{{{x}^{2}}+5}} \\
& =\frac{\left( 2x\sqrt{{{x}^{2}}+5}\cdot \sqrt{{{x}^{2}}+5}-\frac{2{{x}^{3}}}{\sqrt{{{x}^{2}}+5}}\cdot \sqrt{{{x}^{2}}+5} \right)}{{{x}^{2}}+5\cdot \sqrt{{{x}^{2}}+5}} \\
& =\frac{2x\left( {{x}^{2}}+5 \right)-2{{x}^{3}}}{\sqrt{{{\left( {{x}^{2}}+5 \right)}^{3}}}}
\end{align}$
On further simplification:
$\begin{align}
& \frac{2x\sqrt{{{x}^{2}}+5}-\frac{2{{x}^{3}}}{\sqrt{{{x}^{2}}+5}}}{{{x}^{2}}+5}=\frac{2{{x}^{3}}+10x-2{{x}^{3}}}{\sqrt{{{\left( {{x}^{2}}+5 \right)}^{3}}}} \\
& =\frac{10x}{\sqrt{{{\left( {{x}^{2}}+5 \right)}^{3}}}}
\end{align}$
Therefore, the simplified form of the expression $\frac{2x\sqrt{{{x}^{2}}+5}-\frac{2{{x}^{3}}}{\sqrt{{{x}^{2}}+5}}}{{{x}^{2}}+5}$ is $\frac{10x}{\sqrt{{{\left( {{x}^{2}}+5 \right)}^{3}}}}$.
