$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}-2;}{\phantom{;}3x^{4}+6x^{3}+16x^{2}+32x\phantom{;}+69\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}-2\overline{\smash{)}\phantom{;}3x^{5}\phantom{-;x^n}+4x^{3}\phantom{-;x^n}+5x\phantom{;}-12\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}-2;}\underline{-3x^{5}+6x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{5}+6x^{4};}\phantom{;}6x^{4}+4x^{3}\phantom{-;x^n}+5x\phantom{;}-12\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n;}\underline{-6x^{4}+12x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-6x^{4}+12x^{3}-;x^n;}\phantom{;}16x^{3}\phantom{-;x^n}+5x\phantom{;}-12\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n;}\underline{-16x^{3}+32x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-16x^{3}+32x^{2}-;x^n-;x^n;}\phantom{;}32x^{2}+5x\phantom{;}-12\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n;}\underline{-32x^{2}+64x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-32x^{2}+64x\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}69x\phantom{;}-12\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}-2-;x^n-;x^n-;x^n-;x^n;}\underline{-69x\phantom{;}+138\phantom{;}\phantom{;}}\\\phantom{;;;;-69x\phantom{;}+138\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}126\phantom{;}\phantom{;}\\\end{array}$

$3x^{4}+6x^{3}+16x^{2}+32x+69+\frac{126}{x-2}$