$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+2;}{\phantom{;}x^{4}+x^{3}-8x^{2}+18x\phantom{;}-45\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+2\overline{\smash{)}\phantom{;}x^{5}+3x^{4}-6x^{3}+2x^{2}-9x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+2;}\underline{-x^{5}-2x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-2x^{4};}\phantom{;}x^{4}-6x^{3}+2x^{2}-9x\phantom{;}+6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n;}\underline{-x^{4}-2x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-x^{4}-2x^{3}-;x^n;}-8x^{3}+2x^{2}-9x\phantom{;}+6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n;}\underline{\phantom{;}8x^{3}+16x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;\phantom{;}8x^{3}+16x^{2}-;x^n-;x^n;}\phantom{;}18x^{2}-9x\phantom{;}+6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n;}\underline{-18x^{2}-36x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-18x^{2}-36x\phantom{;}-;x^n-;x^n-;x^n;}-45x\phantom{;}+6\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}45x\phantom{;}+90\phantom{;}\phantom{;}}\\\phantom{;;;;\phantom{;}45x\phantom{;}+90\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}96\phantom{;}\phantom{;}\\\end{array}$

$x^{4}+x^{3}-8x^{2}+18x-45+\frac{96}{x+2}$