The least common denominator is $\displaystyle{\left({x}-{3}\right)}{\left({x}+{2}\right)}\usepackage{cmbright}\Huge\$ . So:

$\displaystyle\frac{{6}}{{{x}-{3}}}-\frac{{9}}{{{x}+{2}}}=\frac{{{6}{\left({x}+{2}\right)}}}{{{\left({x}-{3}\right)}{\left({x}+{2}\right)}}}-\frac{{{9}{\left({x}-{3}\right)}}}{{{\left({x}+{2}\right)}{\left({x}-{3}\right)}}}=\frac{{{6}{x}+{12}}}{{{{x}}^{{2}}-{x}-{6}}}-\frac{{{9}{x}-{27}}}{{{{x}}^{{2}}-{x}-{6}}}\usepackage{cmbright}\Huge\$

$\displaystyle=\frac{{{6}{x}+{12}-{9}{x}+{27}}}{{{{x}}^{{2}}-{x}-{6}}}=\frac{{-{3}{x}+{39}}}{{{{x}}^{{2}}-{x}-{6}}}\usepackage{cmbright}\Huge\$