$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+2;}{\phantom{;}4x^{4}-11x^{3}+22x^{2}-45x\phantom{;}+90\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+2\overline{\smash{)}\phantom{;}4x^{5}-3x^{4}\phantom{-;x^n}-x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+2;}\underline{-4x^{5}-8x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-4x^{5}-8x^{4};}-11x^{4}\phantom{-;x^n}-x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n;}\underline{\phantom{;}11x^{4}+22x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}11x^{4}+22x^{3}-;x^n;}\phantom{;}22x^{3}-x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n;}\underline{-22x^{3}-44x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-22x^{3}-44x^{2}-;x^n-;x^n;}-45x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n;}\underline{\phantom{;}45x^{2}+90x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}45x^{2}+90x\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}90x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n-;x^n;}\underline{-90x\phantom{;}-180\phantom{;}\phantom{;}}\\\phantom{;;;;-90x\phantom{;}-180\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}-181\phantom{;}\phantom{;}\\\end{array}$

$4x^{4}-11x^{3}+22x^{2}-45x+90+\frac{-181}{x+2}$