I know only the following facts about binary quadratic forms: for a non-square determinant $D$ with $D \equiv 1$ or $0$ mod 4 all equivalence classes of forms with discriminant $D$ have a reduced form with coeffiecients $a,b,c$ such that $|b| \leq |a| \leq |c|$. For $D<0$, $|a| \leq \sqrt{|D|/3}$. Whereas for $D>0$ we have $|a| \leq \frac{1}{2} \sqrt{D}$. Which gives a bound for $b$ using the very first inequality. This in turn, allows $c$ to be determined, through the discriminant formula, once $a,b$ are chosen.
As for my specific question: for $h(5)$ resp. $h(-8)$. I believe there is only one equivalence class (that of $f(x,y)=x^2+xy-y^2$ resp. $x^2+2y^2$).
Is there any clever, elementary way to verify my claim other than brute force checking all forms whose coefficients satisfy the inequalities and then reducing and comparing the reduced forms?