$\begin{array}{l}\phantom{\phantom{;}2x^{2}-2x\phantom{;}-1;}{\frac{1}{2}x^{2}-\frac{1}{2}x\phantom{;}+\frac{-\frac{1}{2}}{2}\phantom{;}\phantom{;}}\\\phantom{;}2x^{2}-2x\phantom{;}-1\overline{\smash{)}\phantom{;}x^{4}-2x^{3}\phantom{-;x^n}-2x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}2x^{2}-2x\phantom{;}-1;}\underline{-x^{4}+x^{3}+\frac{1}{2}x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{4}+x^{3}+\frac{1}{2}x^{2};}-x^{3}+\frac{1}{2}x^{2}-2x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}-2x\phantom{;}-1-;x^n;}\underline{\phantom{;}x^{3}-x^{2}-\frac{1}{2}x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{3}-x^{2}-\frac{1}{2}x\phantom{;}-;x^n;}-\frac{1}{2}x^{2}-\frac{5}{2}x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}-2x\phantom{;}-1-;x^n-;x^n;}\underline{\phantom{;}0.5x^{2}-0.5x\phantom{;}-\frac{1}{4}\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}0.5x^{2}-0.5x\phantom{;}-\frac{1}{4}\phantom{;}\phantom{;}-;x^n-;x^n;}-3x\phantom{;}-\frac{5}{4}\phantom{;}\phantom{;}\\\end{array}$

$\frac{1}{2}x^{2}-\frac{1}{2}x-\frac{1}{4}+\frac{-3x-\frac{5}{4}}{2x^2-2x-1}$