Prove that

Let us assume that $4-5 \sqrt{2}$ is rational . Then, there exist positive co primes $a$ and $b$ such that

$4-5 \sqrt{2}=\frac{a}{b}$

$5 \sqrt{2}=\frac{a}{b}-4$

$\sqrt{2}=\frac{\frac{a}{b}-4}{5}$

$\sqrt{2}=\frac{a-4 b}{5 b}$

This contradicts the fact that $\sqrt{2}$ is an irrational

Hence $4-5 \sqrt{2}$ is irrational