$\begin{array}{l}\phantom{\phantom{;}2x^{2}-3;}{\phantom{;}2x^{2}-x\phantom{;}+\frac{3}{2}\phantom{;}\phantom{;}}\\\phantom{;}2x^{2}-3\overline{\smash{)}\phantom{;}4x^{4}-2x^{3}-3x^{2}-5x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}2x^{2}-3;}\underline{-4x^{4}\phantom{-;x^n}+6x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-4x^{4}+6x^{2};}-2x^{3}+3x^{2}-5x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}-3-;x^n;}\underline{\phantom{;}2x^{3}\phantom{-;x^n}-3x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}2x^{3}-3x\phantom{;}-;x^n;}\phantom{;}3x^{2}-8x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}-3-;x^n-;x^n;}\underline{-3x^{2}\phantom{-;x^n}+\frac{9}{2}\phantom{;}\phantom{;}}\\\phantom{;;-3x^{2}+\frac{9}{2}\phantom{;}\phantom{;}-;x^n-;x^n;}-8x\phantom{;}+\frac{7}{2}\phantom{;}\phantom{;}\\\end{array}$

$2x^{2}-x+\frac{3}{2}+\frac{-8x+\frac{7}{2}}{2x^2-3}$