$6x^2-5xy-6y^2$ reducible over $\mathbb Z$

Question

Why is $f = 6x^2 -5xy -6y^2$ reducible over $\mathbb Z$?

And: Why is every quadratic form with discriminant $0$ properly equivalent to a unique form $f=ax^2$?

Answer 1

Set $t=\frac xy$ and dehomogeneise $f$:

$f(x,y)=y^2(6t^2-5t+6)$.

Now the single variable $6t^2-5t+6$ has rational roots $-2/3$ and $3/2$, whent the factorisation $$6t^2-5t+6=(3t+2)(2t-3),$$ and finally, rehomogeneising, $$f(x,y)=y(3t+2)(2t-3)=(3ty+2y)(2ty-3y)=(3x+2y)(2x-3y).$$