$\begin{array}{l}\phantom{\phantom{;}2x^{2}+4x\phantom{;}+5;}{\phantom{;}3x^{3}-6x^{2}+\frac{13}{2}x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}2x^{2}+4x\phantom{;}+5\overline{\smash{)}\phantom{;}6x^{5}\phantom{-;x^n}+4x^{3}\phantom{-;x^n}\phantom{-;x^n}-3\phantom{;}\phantom{;}}\\\phantom{\phantom{;}2x^{2}+4x\phantom{;}+5;}\underline{-6x^{5}-12x^{4}-15x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-6x^{5}-12x^{4}-15x^{3};}-12x^{4}-11x^{3}\phantom{-;x^n}\phantom{-;x^n}-3\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}+4x\phantom{;}+5-;x^n;}\underline{\phantom{;}12x^{4}+24x^{3}+30x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}12x^{4}+24x^{3}+30x^{2}-;x^n;}\phantom{;}13x^{3}+30x^{2}\phantom{-;x^n}-3\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}+4x\phantom{;}+5-;x^n-;x^n;}\underline{-13x^{3}-26x^{2}-\frac{65}{2}x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-13x^{3}-26x^{2}-\frac{65}{2}x\phantom{;}-;x^n-;x^n;}\phantom{;}4x^{2}-\frac{65}{2}x\phantom{;}-3\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x^{2}+4x\phantom{;}+5-;x^n-;x^n-;x^n;}\underline{-4x^{2}-8x\phantom{;}-10\phantom{;}\phantom{;}}\\\phantom{;;;-4x^{2}-8x\phantom{;}-10\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}-\frac{81}{2}x\phantom{;}-13\phantom{;}\phantom{;}\\\end{array}$

$3x^{3}-6x^{2}+\frac{13}{2}x+2+\frac{-\frac{81}{2}x-13}{2x^2+4x+5}$