3√2 is an irrational number and subtraction of a rational number and an irrational number is an irrational number.

∴ 4 + 3√2 is irrational, where 3√2 is an irrational number while 4 is a rational number.

Alternative - Assume that 4 + 3√2 is rational
4 + 3√2 = a/b, where a and b are integers.
⇒ 3√2 = a/b - 4
⇒ 3√2 = a/b – 4b/b
⇒ √2 = (a – 4b) / 3b
we know that a, b, 3 and 4 are integers and they are also rational {i.e RHS is rational}
therefore √2 will be rational.
but we know that √2 is irrational.
there is a contradiction
so, 4 + 3√2 is an irrational number