I calculated the Gaussian curvature of a metric and obtained the following expression $$ \kappa(r, Q) = \frac{2 \chi E^2 r^2 Q^2 \bigl[ r^2 \xi \!+\! m^2 Q^2 (r^2 \!+\! 2Q^2) \!+\! r^4(E^2\!-\!m^2) \bigr] \!+\! \zeta \xi (r^2 \!+\! Q^2) \bigl[ (r^2 \!+\! Q^2) \xi \!+\! E^2 r^2 Q^2\bigr]\!\!} {2 r^4(r^2 \!+\! Q^2) \xi^3} $$ where \begin{align} \xi &= E^2 r^2 - m^2 (r^2 + Q^2) \\ \chi &= r^2 - rb + Q^2 \\ \zeta &= r(rb' - b) + 2Q^2. \end{align}
where the primed symbol means derivative with respect to $r$, i.e. $b'(r) = \frac{d b(r)}{d r}$, $b(r) = 1 / r^{q}$ para $q>0$ y que $E > 0$.
I understand that the sign of the curvature will depend not only on $r$ and $Q$ but also on $b$, $b'$, and $m$. Is there any technique or way to determine the sign of the Gaussian curvature of expressions this complicated? Could anyone give me a suggestion or recommendation? I searched around on some articles without success.