It appears that you have made some sign errors when subtracting. Start with:

$\displaystyle{f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}={{x}}^{{3}}-{2}{x}+{6}-{\left({3}{{x}}^{{3}}-{2}{{x}}^{{2}}+{4}{x}-{2}\right)}\usepackage{cmbright}\Huge\$

Be sure to distribute the minus sign to all terms of the second function and then combine like terms.