Answer
$$ - \frac{3}{{5{x^2}}} + \frac{3}{{5\left( {{x^2} + 5} \right)}}$$
Work Step by Step
$$\eqalign{
& \frac{{ - 3}}{{{x^2}\left( {{x^2} + 5} \right)}} \cr
& {\text{The partial fraction decomposition is }} \cr
& \frac{{ - 3}}{{{x^2}\left( {{x^2} + 5} \right)}} = \frac{A}{x} + \frac{B}{{{x^2}}} + \frac{{Cx + D}}{{{x^2} + 5}} \cr
& {\text{Multiply each side by }}{x^2}\left( {{x^2} + 5} \right) \cr
& - 3 = Ax\left( {{x^2} + 5} \right) + B\left( {{x^2} + 5} \right) + \left( {Cx + D} \right){x^2}\,\,\,\,\,\,\,\,\left( 1 \right) \cr
& \cr
& {\text{Expand and combine like terms on the right of }}\left( 1 \right) \cr
& - 3 = A{x^3} + 5Ax + B{x^2} + 5B + C{x^3} + D{x^2}\, \cr
& - 3 = \left( {A{x^3} + C{x^3}\,} \right) + \left( {B{x^2} + D{x^2}} \right) + 5Ax + 5B \cr
& {\text{Equating the coefficients}} \cr
& 5B = - 3,\,\,\,\,\,B = - \frac{3}{5} \cr
& B + D = 0,\,\,\,\,D = \frac{3}{5} \cr
& 5A = 0,\,\,\,A = 0 \cr
& A + C = 0,\,\,C = 0 \cr
& \cr
& {\text{Use }}A,{\text{ }}B{\text{ and }}C{\text{ to find the partial fraction decomposition}} \cr
& \frac{{ - 3}}{{{x^2}\left( {{x^2} + 5} \right)}} = \frac{0}{x} + \frac{{ - 3/5}}{{{x^2}}} + \frac{{3/5}}{{{x^2} + 5}} \cr
& \frac{{ - 3}}{{{x^2}\left( {{x^2} + 5} \right)}} = - \frac{3}{{5{x^2}}} + \frac{3}{{5\left( {{x^2} + 5} \right)}} \cr} $$
