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

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