$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+2;}{-5x^{3}+19x^{2}-30x\phantom{;}+65\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+2\overline{\smash{)}-5x^{4}+9x^{3}+8x^{2}+5x\phantom{;}+5\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+2;}\underline{\phantom{;}5x^{4}+10x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}5x^{4}+10x^{3};}\phantom{;}19x^{3}+8x^{2}+5x\phantom{;}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n;}\underline{-19x^{3}-38x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-19x^{3}-38x^{2}-;x^n;}-30x^{2}+5x\phantom{;}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n;}\underline{\phantom{;}30x^{2}+60x\phantom{;}\phantom{-;x^n}}\\\phantom{;;\phantom{;}30x^{2}+60x\phantom{;}-;x^n-;x^n;}\phantom{;}65x\phantom{;}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n;}\underline{-65x\phantom{;}-130\phantom{;}\phantom{;}}\\\phantom{;;;-65x\phantom{;}-130\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}-125\phantom{;}\phantom{;}\\\end{array}$

$-5x^{3}+19x^{2}-30x+65+\frac{-125}{2+x}$