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

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