5√2 is an irrational number and subtraction of a rational number and an irrational number is an irrational number.

∴, 4 - 5√2 is irrational, where 5√2 is an irrational number while 4 is a rational number.

Alternative - Assume that 4 - 5√2 is rational
4 - 5√2 = a/b , where a and b are integers .
⇒ - 5√2 = a/b - 4
⇒ 5√2 = 4 - a/b
⇒ 5√2 = 4b/b - a/b
⇒ √2 = (4b – a) / 5b
we know that a, b, 4 and 5 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 - 5√2 is an irrational number