$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+2;}{\phantom{;}5x^{3}-10x^{2}+12x\phantom{;}-22\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+2\overline{\smash{)}\phantom{;}5x^{4}\phantom{-;x^n}-8x^{2}+2x\phantom{;}-40\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+2;}\underline{-5x^{4}-10x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-5x^{4}-10x^{3};}-10x^{3}-8x^{2}+2x\phantom{;}-40\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n;}\underline{\phantom{;}10x^{3}+20x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}10x^{3}+20x^{2}-;x^n;}\phantom{;}12x^{2}+2x\phantom{;}-40\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n;}\underline{-12x^{2}-24x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-12x^{2}-24x\phantom{;}-;x^n-;x^n;}-22x\phantom{;}-40\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n;}\underline{\phantom{;}22x\phantom{;}+44\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}22x\phantom{;}+44\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}4\phantom{;}\phantom{;}\\\end{array}$

$5x^{3}-10x^{2}+12x-22+\frac{4}{x+2}$