Hi i hope this answer helps you!
step 1: 3 + 2 sqrt(5) is rational if and only if sqrt(5) is rational. This should be easy for you to show.
step 2: we now show sqrt(5) is irrational. Suppose sqrt(5) = a/b. The 5b^2 = a^2. By unique factorization theorem, the both a and b have prime factorisations, say b = p_1^e_1*...*p_k^e_k and a = q_1^f_1 * .. * q_l^f_l , where the p_i are unique, same with q_j. The squaring results in doubling the exponents, so the right hand side has all even exponents. On the other hand, the left hand side has all even exponents except for the prime 5 which has odd. By unique factorization, this cannot happen: we cannot have an odd exponent on one side and all even exponents on the other. Therefore, it cannot be rational.
