$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+2;}{\phantom{;}3x^{5}-6x^{4}+12x^{3}-31x^{2}+62x\phantom{;}-123\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+2\overline{\smash{)}\phantom{;}3x^{6}\phantom{-;x^n}\phantom{-;x^n}-7x^{3}\phantom{-;x^n}+x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+2;}\underline{-3x^{6}-6x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{6}-6x^{5};}-6x^{5}\phantom{-;x^n}-7x^{3}\phantom{-;x^n}+x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n;}\underline{\phantom{;}6x^{5}+12x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}6x^{5}+12x^{4}-;x^n;}\phantom{;}12x^{4}-7x^{3}\phantom{-;x^n}+x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n;}\underline{-12x^{4}-24x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-12x^{4}-24x^{3}-;x^n-;x^n;}-31x^{3}\phantom{-;x^n}+x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n;}\underline{\phantom{;}31x^{3}+62x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}31x^{3}+62x^{2}-;x^n-;x^n-;x^n;}\phantom{;}62x^{2}+x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n-;x^n;}\underline{-62x^{2}-124x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;-62x^{2}-124x\phantom{;}-;x^n-;x^n-;x^n-;x^n;}-123x\phantom{;}-1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+2-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}123x\phantom{;}+246\phantom{;}\phantom{;}}\\\phantom{;;;;;\phantom{;}123x\phantom{;}+246\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}245\phantom{;}\phantom{;}\\\end{array}$

$3x^{5}-6x^{4}+12x^{3}-31x^{2}+62x-123+\frac{245}{x+2}$