You may not have distributed the subtraction sign to all the terms in the function $\displaystyle{g{{\left({x}\right)}}}\usepackage{cmbright}\Huge\$ . Set up the problem as follows:

$\displaystyle{\left(\frac{{{h}-{g}}}{{f}}\right)}=\frac{{{h}{\left({x}\right)}-{g{{\left({x}\right)}}}}}{{{f{{\left({x}\right)}}}}}=\frac{{{12}{{x}}^{{3}}+{6}{{x}}^{{2}}+{20}-{\left(-{12}{{x}}^{{2}}+{6}{x}+{20}\right)}}}{{{6}{x}}}\usepackage{cmbright}\Huge\$

$\displaystyle=\frac{{{12}{{x}}^{{3}}+{6}{{x}}^{{2}}+{20}+{12}{{x}}^{{2}}-{6}{x}-{20}}}{{{6}{x}}}\usepackage{cmbright}\Huge\$

Combine like terms, factor the numerator, and simplify.