We have $$3x^2+ 5xy -2y^2 = 12.$$
Prove the major axis lies on $y = \tan(22.5)x$.
The hyperbola is a rotation of $22.5$ degrees of a normal hyperbola, which I think is why the slope of the major axis is $\tan(22.5)$.
I don't know how to prove this though.
Edit:
I figured out that the original equation can be rewritten as $x^2/4 + 5xy/12 - y^2/6 = 1$, so the non rotated hyperbola is $x^2/4 - y^2/6 = 1$.
$$ 3x^2 + 5xy - 2y^2 = 0\\ (3x - y)(x + 2y) = 0\\ $$
So the lines $y = 3x$ and $y = \frac{-1}{2} x$ are asymptotes. Bisect these by finding the angles each make with the positive x-axis.