First we, have to prove that √5 is an irrational number.

Let us suppose that √5 is a rational number.

So √5 = p/q

=> 5 = p² /q²

=>5q² = p² ..............1

So p² is divisible by 5.

=> p is divisible by 5.

Let p =5x  (x is a positive integer)

Now p²  = 25c²

from equation 1

5q² = 25c²

=> q² = 5c²

So q is divisible by 5

Thus p and q has a common factor 5. It is contradiction of our assumption.

So, √5 is not a rational number.

Therefore, √5 is an irrational number.

Since, √5 is an irrational number.

Hence, 3 + 2√5 is also an irrational number.