Let us assume, for contradiction, that $ 3 + 2\sqrt{5} $ is rational.
That means we can write it as:
$ 3 + 2\sqrt{5} = \frac{a}{b} $, where $ a, b $ are integers and $ b \neq 0 $.
Rearranging:
$ 2\sqrt{5} = \frac{a}{b} - 3 $
$ 2\sqrt{5} = \frac{a - 3b}{b} $
Dividing by 2:
$ \sqrt{5} = \frac{a - 3b}{2b} $
Since $ a $ and $ b $ are integers, $ \frac{a - 3b}{2b} $ is a rational number.
However, $ \sqrt{5} $ is known to be irrational, which contradicts our assumption.
Therefore, $ 3 + 2\sqrt{5} $ is irrational.