Assume that $3+2\sqrt{5}$ is rational. Then,

$3+2\sqrt{5},=\frac{\mathrm{a}}{\mathrm{b}}$

$\sqrt{5},=\frac{\mathrm{a}-3\mathrm{b}}{2\mathrm{b}}$

This gives that $\frac{\mathrm{a}-3\mathrm{b}}{2\mathrm{b}}$ is rational but $\sqrt{5}$ is an irrational number, therefore, the assumption is wrong.

Hence, $3+2\sqrt{5}$ is irrational.