1. \(\displaystyle\frac{{{x}^{{4}}+{6}}}{{{x}^{{5}}+{7}{x}^{{3}}}}=\frac{{{x}^{{4}}+{6}}}{{{x}^{{3}}{\left({x}^{{2}}+{7}\right)}}}\)

The partial fraction:

\(\displaystyle=\frac{{A}}{{x}}+\frac{{B}}{{x}^{{2}}}+\frac{{C}}{{x}^{{3}}}+\frac{{{D}{x}+{E}}}{{{x}^{{2}}+{7}}}\)

Where A-E are constants.

2. \(\displaystyle\frac{{2}}{{{\left({x}^{{2}}-{9}\right)}^{{2}}}}=\frac{{2}}{{{\left({\left({x}+{3}\right)}{\left({x}-{3}\right)}\right)}^{{2}}}}\)

\(\displaystyle=\frac{{2}}{{{\left({x}+{3}\right)}^{{2}}{\left({x}-{3}\right)}^{{2}}}}\)

The partial fraction:

\(\displaystyle=\frac{{A}}{{{x}+{3}}}+\frac{{B}}{{{\left({x}+{3}\right)}^{{2}}}}+\frac{{C}}{{{x}-{3}}}+\frac{{D}}{{{\left({x}-{3}\right)}^{{2}}}}\)

Where A-D are constants.

The partial fraction:

\(\displaystyle=\frac{{A}}{{x}}+\frac{{B}}{{x}^{{2}}}+\frac{{C}}{{x}^{{3}}}+\frac{{{D}{x}+{E}}}{{{x}^{{2}}+{7}}}\)

Where A-E are constants.

2. \(\displaystyle\frac{{2}}{{{\left({x}^{{2}}-{9}\right)}^{{2}}}}=\frac{{2}}{{{\left({\left({x}+{3}\right)}{\left({x}-{3}\right)}\right)}^{{2}}}}\)

\(\displaystyle=\frac{{2}}{{{\left({x}+{3}\right)}^{{2}}{\left({x}-{3}\right)}^{{2}}}}\)

The partial fraction:

\(\displaystyle=\frac{{A}}{{{x}+{3}}}+\frac{{B}}{{{\left({x}+{3}\right)}^{{2}}}}+\frac{{C}}{{{x}-{3}}}+\frac{{D}}{{{\left({x}-{3}\right)}^{{2}}}}\)

Where A-D are constants.