Answer
The partial fraction expansion is $\frac{{{x}^{3}}+{{x}^{2}}}{{{\left( {{x}^{2}}+4 \right)}^{2}}}=\frac{Ax+B}{{{x}^{2}}+4}+\frac{Cx+D}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$.
Work Step by Step
The provided rational expression is as given below:
$\frac{{{x}^{3}}+{{x}^{2}}}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$
Now, solve the expression as follows:
We set up the partial fraction expansion with unknown constants coefficients and then write a constant coefficients over each of the two distinct algebraic linear factors in the denominator of the expression.
Then, decompose the fractional part as follows:
$\frac{{{x}^{3}}+{{x}^{2}}}{{{\left( {{x}^{2}}+4 \right)}^{2}}}=\frac{Ax+B}{{{x}^{2}}+4}+\frac{Cx+D}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$
Thus, $\frac{Ax+B}{{{x}^{2}}+4}+\frac{Cx+D}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$ is a partial fraction expansion of rational expression $\frac{{{x}^{3}}+{{x}^{2}}}{{{\left( {{x}^{2}}+4 \right)}^{2}}}$ with constants $A$,$B$,$C$ and $D$.
