$\begin{array}{l}\phantom{\phantom{;}x^{2}-3x\phantom{;}+1;}{\phantom{;}x^{4}+3x^{3}+8x^{2}+21x\phantom{;}+52\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-3x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-3x^{2}+5x\phantom{;}-8\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-3x\phantom{;}+1;}\underline{-x^{6}+3x^{5}-x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{6}+3x^{5}-x^{4};}\phantom{;}3x^{5}-x^{4}\phantom{-;x^n}-3x^{2}+5x\phantom{;}-8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-3x\phantom{;}+1-;x^n;}\underline{-3x^{5}+9x^{4}-3x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-3x^{5}+9x^{4}-3x^{3}-;x^n;}\phantom{;}8x^{4}-3x^{3}-3x^{2}+5x\phantom{;}-8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-3x\phantom{;}+1-;x^n-;x^n;}\underline{-8x^{4}+24x^{3}-8x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-8x^{4}+24x^{3}-8x^{2}-;x^n-;x^n;}\phantom{;}21x^{3}-11x^{2}+5x\phantom{;}-8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-3x\phantom{;}+1-;x^n-;x^n-;x^n;}\underline{-21x^{3}+63x^{2}-21x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;-21x^{3}+63x^{2}-21x\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}52x^{2}-16x\phantom{;}-8\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-3x\phantom{;}+1-;x^n-;x^n-;x^n-;x^n;}\underline{-52x^{2}+156x\phantom{;}-52\phantom{;}\phantom{;}}\\\phantom{;;;;-52x^{2}+156x\phantom{;}-52\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}140x\phantom{;}-60\phantom{;}\phantom{;}\\\end{array}$

$x^{4}+3x^{3}+8x^{2}+21x+52+\frac{140x-60}{x^2-3x+1}$