$\begin{array}{l}\phantom{\phantom{;}4x^{2}+2x\phantom{;}-3;}{\phantom{;}3x^{2}-x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}4x^{2}+2x\phantom{;}-3\overline{\smash{)}\phantom{;}12x^{4}+2x^{3}-3x^{2}+12x\phantom{;}-9\phantom{;}\phantom{;}}\\\phantom{\phantom{;}4x^{2}+2x\phantom{;}-3;}\underline{-12x^{4}-6x^{3}+9x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-12x^{4}-6x^{3}+9x^{2};}-4x^{3}+6x^{2}+12x\phantom{;}-9\phantom{;}\phantom{;}\\\phantom{\phantom{;}4x^{2}+2x\phantom{;}-3-;x^n;}\underline{\phantom{;}4x^{3}+2x^{2}-3x\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}4x^{3}+2x^{2}-3x\phantom{;}-;x^n;}\phantom{;}8x^{2}+9x\phantom{;}-9\phantom{;}\phantom{;}\\\phantom{\phantom{;}4x^{2}+2x\phantom{;}-3-;x^n-;x^n;}\underline{-8x^{2}-4x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{;;-8x^{2}-4x\phantom{;}+6\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}5x\phantom{;}-3\phantom{;}\phantom{;}\\\end{array}$

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