$\begin{array}{l}\phantom{\phantom{;}r^{2}-1;}{-2r^{4}+r^{3}-2r^{2}+r\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{;}r^{2}-1\overline{\smash{)}-2r^{6}+r^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-r\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}r^{2}-1;}\underline{\phantom{;}2r^{6}\phantom{-;x^n}-2r^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}2r^{6}-2r^{4};}\phantom{;}r^{5}-2r^{4}\phantom{-;x^n}\phantom{-;x^n}-r\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}r^{2}-1-;x^n;}\underline{-r^{5}\phantom{-;x^n}+r^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-r^{5}+r^{3}-;x^n;}-2r^{4}+r^{3}\phantom{-;x^n}-r\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}r^{2}-1-;x^n-;x^n;}\underline{\phantom{;}2r^{4}\phantom{-;x^n}-2r^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;\phantom{;}2r^{4}-2r^{2}-;x^n-;x^n;}\phantom{;}r^{3}-2r^{2}-r\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}r^{2}-1-;x^n-;x^n-;x^n;}\underline{-r^{3}\phantom{-;x^n}+r\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-r^{3}+r\phantom{;}-;x^n-;x^n-;x^n;}-2r^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}r^{2}-1-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}2r^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{;;;;\phantom{;}2r^{2}-2\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}-1\phantom{;}\phantom{;}\\\end{array}$

$-2r^{4}+r^{3}-2r^{2}+r-2+\frac{-1}{r^2-1}$