$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+2;}{-5x^{3}+7x^{2}-12x\phantom{;}+22\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+2\overline{\smash{)}-5x^{4}-3x^{3}+2x^{2}-2x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+2;}\underline{\phantom{;}5x^{4}+10x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}5x^{4}+10x^{3};}\phantom{;}7x^{3}+2x^{2}-2x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n;}\underline{-7x^{3}-14x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-7x^{3}-14x^{2}-;x^n;}-12x^{2}-2x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n;}\underline{\phantom{;}12x^{2}+24x\phantom{;}\phantom{-;x^n}}\\\phantom{;;\phantom{;}12x^{2}+24x\phantom{;}-;x^n-;x^n;}\phantom{;}22x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n;}\underline{-22x\phantom{;}-44\phantom{;}\phantom{;}}\\\phantom{;;;-22x\phantom{;}-44\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}-43\phantom{;}\phantom{;}\\\end{array}$

$-5x^{3}+7x^{2}-12x+22+\frac{-43}{x+2}$