You may have correctly rewritten each fraction with the least common multiple (LCM) as the denominator:

$\displaystyle\frac{{{3}{x}-{2}}}{{{x}-{2}}}-\frac{{{x}-{1}}}{{{{x}}^{{2}}-{5}{x}+{6}}}=\frac{{{\left({3}{x}-{2}\right)}{\left({x}-{3}\right)}}}{{{\left({x}-{2}\right)}{\left({x}-{3}\right)}}}-\frac{{{x}-{1}}}{{{\left({x}-{2}\right)}{\left({x}-{3}\right)}}}\usepackage{cmbright}\Huge\$

Here you may have incorrectly multiplied $\displaystyle{\left({3}{x}-{2}\right)}\usepackage{cmbright}\Huge\$ by $\displaystyle{\left({x}-{3}\right)}\usepackage{cmbright}\Huge\$ , obtaining $\displaystyle{\left({3}{{x}}^{{2}}+{6}\right)}\usepackage{cmbright}\Huge\$ . Remember to use FOIL.