$\begin{array}{l}\phantom{\phantom{;}t\phantom{;}+3;}{\phantom{;}4t^{3}-12t^{2}+36t\phantom{;}-108\phantom{;}\phantom{;}}\\\phantom{;}t\phantom{;}+3\overline{\smash{)}\phantom{;}4t^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}t\phantom{;}+3;}\underline{-4t^{4}-12t^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-4t^{4}-12t^{3};}-12t^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}t\phantom{;}+3-;x^n;}\underline{\phantom{;}12t^{3}+36t^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}12t^{3}+36t^{2}-;x^n;}\phantom{;}36t^{2}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}t\phantom{;}+3-;x^n-;x^n;}\underline{-36t^{2}-108t\phantom{;}\phantom{-;x^n}}\\\phantom{;;-36t^{2}-108t\phantom{;}-;x^n-;x^n;}-108t\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}t\phantom{;}+3-;x^n-;x^n-;x^n;}\underline{\phantom{;}108t\phantom{;}+324\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}108t\phantom{;}+324\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}324\phantom{;}\phantom{;}\\\end{array}$

$4t^{3}-12t^{2}+36t-108+\frac{324}{t+3}$