$\begin{array}{l}\phantom{-x^{2}+3;}{\phantom{;}x^{10}\phantom{-;x^n}+3x^{8}\phantom{-;x^n}+9x^{6}\phantom{-;x^n}+27x^{4}\phantom{-;x^n}+81x^{2}\phantom{-;x^n}+243\phantom{;}\phantom{;}}\\-x^{2}+3\overline{\smash{)}-x^{12}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+81\phantom{;}\phantom{;}}\\\phantom{-x^{2}+3;}\underline{\phantom{;}x^{12}\phantom{-;x^n}-3x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}x^{12}-3x^{10};}-3x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+81\phantom{;}\phantom{;}\\\phantom{-x^{2}+3-;x^n;}\underline{\phantom{;}3x^{10}\phantom{-;x^n}-9x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}3x^{10}-9x^{8}-;x^n;}-9x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+81\phantom{;}\phantom{;}\\\phantom{-x^{2}+3-;x^n-;x^n;}\underline{\phantom{;}9x^{8}\phantom{-;x^n}-27x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;\phantom{;}9x^{8}-27x^{6}-;x^n-;x^n;}-27x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+81\phantom{;}\phantom{;}\\\phantom{-x^{2}+3-;x^n-;x^n-;x^n;}\underline{\phantom{;}27x^{6}\phantom{-;x^n}-81x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}27x^{6}-81x^{4}-;x^n-;x^n-;x^n;}-81x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+81\phantom{;}\phantom{;}\\\phantom{-x^{2}+3-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}81x^{4}\phantom{-;x^n}-243x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;\phantom{;}81x^{4}-243x^{2}-;x^n-;x^n-;x^n-;x^n;}-243x^{2}\phantom{-;x^n}+81\phantom{;}\phantom{;}\\\phantom{-x^{2}+3-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}243x^{2}\phantom{-;x^n}-729\phantom{;}\phantom{;}}\\\phantom{;;;;;\phantom{;}243x^{2}-729\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n;}-648\phantom{;}\phantom{;}\\\end{array}$

$x^{10}+3x^{8}+9x^{6}+27x^{4}+81x^{2}+243+\frac{-648}{3-x^2}$