$\begin{array}{l}\phantom{\phantom{;}y^{2}-3;}{\phantom{;}32y^{13}\phantom{-;x^n}+96y^{11}\phantom{-;x^n}+288y^{9}\phantom{-;x^n}+864y^{7}\phantom{-;x^n}+2592y^{5}\phantom{-;x^n}+7776y^{3}\phantom{-;x^n}+23328y\phantom{;}\phantom{-;x^n}}\\\phantom{;}y^{2}-3\overline{\smash{)}\phantom{;}32y^{15}\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}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-243\phantom{;}\phantom{;}}\\\phantom{\phantom{;}y^{2}-3;}\underline{-32y^{15}\phantom{-;x^n}+96y^{13}\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}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-32y^{15}+96y^{13};}\phantom{;}96y^{13}\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}\phantom{-;x^n}-243\phantom{;}\phantom{;}\\\phantom{\phantom{;}y^{2}-3-;x^n;}\underline{-96y^{13}\phantom{-;x^n}+288y^{11}\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}}\\\phantom{;-96y^{13}+288y^{11}-;x^n;}\phantom{;}288y^{11}\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}-243\phantom{;}\phantom{;}\\\phantom{\phantom{;}y^{2}-3-;x^n-;x^n;}\underline{-288y^{11}\phantom{-;x^n}+864y^{9}\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{;;-288y^{11}+864y^{9}-;x^n-;x^n;}\phantom{;}864y^{9}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-243\phantom{;}\phantom{;}\\\phantom{\phantom{;}y^{2}-3-;x^n-;x^n-;x^n;}\underline{-864y^{9}\phantom{-;x^n}+2592y^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-864y^{9}+2592y^{7}-;x^n-;x^n-;x^n;}\phantom{;}2592y^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-243\phantom{;}\phantom{;}\\\phantom{\phantom{;}y^{2}-3-;x^n-;x^n-;x^n-;x^n;}\underline{-2592y^{7}\phantom{-;x^n}+7776y^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;-2592y^{7}+7776y^{5}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}7776y^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}-243\phantom{;}\phantom{;}\\\phantom{\phantom{;}y^{2}-3-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-7776y^{5}\phantom{-;x^n}+23328y^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;-7776y^{5}+23328y^{3}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}23328y^{3}\phantom{-;x^n}\phantom{-;x^n}-243\phantom{;}\phantom{;}\\\phantom{\phantom{;}y^{2}-3-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-23328y^{3}\phantom{-;x^n}+69984y\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;;;-23328y^{3}+69984y\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}69984y\phantom{;}-243\phantom{;}\phantom{;}\\\end{array}$

$32y^{13}+96y^{11}+288y^{9}+864y^{7}+2592y^{5}+7776y^{3}+23328y+\frac{69984y-243}{y^2-3}$