I'll do a proof by contradiction.
Assume 5-2*sqrt(3) is rational. If so, then it's of the form p/q for some integers p and q where q is nonzero.
5-2*sqrt(3) = p/q
-2*sqrt(3) = (p/q) - 5
-2*sqrt(3) = (p/q) - (5q)/q
-2*sqrt(3) = (p-5q)/q
sqrt(3) = (p-5q)/(-2q)
The numerator p-5q is some integer, and -2q is some nonzero integer. This means (p-5q)/(-2q) is rational.
But this contradicts sqrt(3) being irrational. This contradiction happens because we made the assumption that 5-2*sqrt(3) is rational.
Therefore 5-2*sqrt(3) must be irrational.