$\begin{array}{l}\phantom{\phantom{;}2x^{2}+x\phantom{;}-2;}{\frac{5}{2}x^{2}+\frac{-\frac{31}{2}}{2}x\phantom{;}+\frac{\frac{51}{4}}{2}\phantom{;}\phantom{;}}\\\phantom{;}2x^{2}+x\phantom{;}-2\overline{\smash{)}\phantom{;}5x^{4}-13x^{3}\phantom{-;x^n}+15x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{\phantom{;}2x^{2}+x\phantom{;}-2;}\underline{-5x^{4}-\frac{5}{2}x^{3}+5x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-5x^{4}-\frac{5}{2}x^{3}+5x^{2};}-\frac{31}{2}x^{3}+5x^{2}+15x\phantom{;}-6\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}+x\phantom{;}-2-;x^n;}\underline{\phantom{;}15.5x^{3}+\frac{31}{4}x^{2}-15.5x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}15.5x^{3}+\frac{31}{4}x^{2}-15.5x\phantom{;}-;x^n;}\frac{51}{4}x^{2}-0.5x\phantom{;}-6\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}+x\phantom{;}-2-;x^n-;x^n;}\underline{-12.75x^{2}-\frac{51}{8}x\phantom{;}+12.75\phantom{;}\phantom{;}}\\\phantom{;;-12.75x^{2}-\frac{51}{8}x\phantom{;}+12.75\phantom{;}\phantom{;}-;x^n-;x^n;}-\frac{55}{8}x\phantom{;}+6.75\phantom{;}\phantom{;}\\\end{array}$

$\frac{5}{2}x^{2}-\frac{31}{4}x+\frac{51}{8}+\frac{-\frac{55}{8}x+6.75}{2x^2+x-2}$