Expand $(3+\sqrt{5})^4$. Enter your answer in the form $a + b\sqrt{5}$ for some positive integers $a$ and $b$.

$\mathrm{Apply\:binomial\:theorem}:\quad \left(a+b\right)^n=\sum _{i=0}^n\binom{n}{i}a^{\left(n-i\right)}b^i$

$\sum _{i=0}^4\binom{4}{i}\cdot \:3^{\left(4-i\right)}\left(\sqrt{5}\right)^i$

$81+108\sqrt{5}+270+60\sqrt{5}+25$

$376+168\sqrt{5}$

-Vinculum