Negative definite quadratic form

Reduce $-Q$.
$$-Q=x^2+2xy+3y^2=(x+y)^2+2y^2.$$

$-Q$ is positive definite. It is a sum of positive squares, and vanishes only when $x+y=0$ and $y=0$, i.e. $(x,y)=(0,0)$.

Conclusion. $Q=-[(x+y)^2+2y^2]$ is negative definite: $Q(x,y)<0$ for $(x,y)\neq(0,0)$, with signature $(0,2,0)$.