$\begin{array}{l}\phantom{\phantom{;}4x^{2}-2x\phantom{;}+1;}{\frac{15}{4}x^{2}-\frac{6.5}{4}x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}4x^{2}-2x\phantom{;}+1\overline{\smash{)}\phantom{;}15x^{4}-14x^{3}+15x^{2}-6x\phantom{;}+4\phantom{;}\phantom{;}}\\\phantom{\phantom{;}4x^{2}-2x\phantom{;}+1;}\underline{-15x^{4}+7.5x^{3}-\frac{15}{4}x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-15x^{4}+7.5x^{3}-\frac{15}{4}x^{2};}-6.5x^{3}+\frac{45}{4}x^{2}-6x\phantom{;}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}4x^{2}-2x\phantom{;}+1-;x^n;}\underline{\phantom{;}6.5x^{3}-3.25x^{2}+1.625x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}6.5x^{3}-3.25x^{2}+1.625x\phantom{;}-;x^n;}\phantom{;}8x^{2}-4.375x\phantom{;}+4\phantom{;}\phantom{;}\\\phantom{\phantom{;}4x^{2}-2x\phantom{;}+1-;x^n-;x^n;}\underline{-8x^{2}+4x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{;;-8x^{2}+4x\phantom{;}-2\phantom{;}\phantom{;}-;x^n-;x^n;}-0.375x\phantom{;}+2\phantom{;}\phantom{;}\\\end{array}$

$\frac{15}{4}x^{2}-\frac{6.5}{4}x+2+\frac{-0.375x+2}{4x^2-2x+1}$