Answer
The simplified form of the equation is $\frac{5{{x}^{3}}-3{{x}^{2}}+7x-3}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$.
Work Step by Step
Let us consider the expression $\frac{5x-3}{{{x}^{2}}+1}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$
And simplify:
$\begin{align}
& \frac{{{\left( {{x}^{2}}+1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\cdot \left( \frac{5x-3}{{{x}^{2}}+1}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \right)=\frac{{{\left( {{x}^{2}}+1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\cdot \frac{5x-3}{{{x}^{2}}+1}+\frac{{{\left( {{x}^{2}}+1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\cdot \frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \\
& =\frac{\left( {{x}^{2}}+1 \right)\left( 5x-3 \right)}{{{\left( {{x}^{2}}+1 \right)}^{2}}}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \\
& =\frac{{{x}^{2}}\left( 5x-3 \right)+1\left( 5x-3 \right)+2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \\
& =\frac{5{{x}^{3}}-3{{x}^{2}}+5x-3+2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}}
\end{align}$
Therefore,
$\frac{{{\left( {{x}^{2}}+1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\cdot \left( \frac{5x-3}{{{x}^{2}}+1}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \right)=\frac{5{{x}^{3}}-3{{x}^{2}}+7x-3}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$
Thus, the simplified form of the expression $\frac{5x-3}{{{x}^{2}}+1}+\frac{2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$ is $\frac{5{{x}^{3}}-3{{x}^{2}}+7x-3}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$.
