Let us assume, to the contrary that 4 - 5√2 is rational.
So, we can find coprime integers a and b(b ≠ 0)
such that 4 - 5√2 = a/b
=> 5√2 = 4 - a/b
=> √2 = (4b - a)/5b
Since a and b are integers, (4b - a)/5b is rational.
So, √2 is rational.
But this contradicts the fact that √2 is irrational.
Hence, 4 - 5√2 is irrational.