$\begin{array}{l}\phantom{\phantom{;}x^{2}+3x\phantom{;}+3;}{\phantom{;}x^{2}-3x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3x\phantom{;}+3\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+3;}\underline{-x^{4}-3x^{3}-3x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}-3x^{3}-3x^{2};}-3x^{3}-3x^{2}\phantom{-;x^n}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+3-;x^n;}\underline{\phantom{;}3x^{3}+9x^{2}+9x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}3x^{3}+9x^{2}+9x\phantom{;}-;x^n;}\phantom{;}6x^{2}+9x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+3-;x^n-;x^n;}\underline{-6x^{2}-18x\phantom{;}-18\phantom{;}\phantom{;}}\\\phantom{;;-6x^{2}-18x\phantom{;}-18\phantom{;}\phantom{;}-;x^n-;x^n;}-9x\phantom{;}-16\phantom{;}\phantom{;}\\\end{array}$

$x^{2}-3x+6+\frac{-9x-16}{x^2+3x+3}$