I have the following expression for \sigma
\sigma = \lambda -15 (\rho_a^2 + \rho_b^2) + \sqrt{9(\rho_a^4 + \rho_b^4) + \gamma^2 - 48 \gamma \rho_a \rho_b + 558 \rho_a^2 \rho_b^2}
And I wish to find the ranges of \lambda for which \sigma is negative. The forms of \rho_a and \rho_b are the following
\rho_a = \sqrt{\frac{\lambda + \sqrt{\lambda^2 - \gamma^2}}{6}} \\\rho_b = \sqrt{\frac{\lambda - \sqrt{\lambda^2 - \gamma^2}}{6}}
When I use Solve[\sigma < 0 , \lambda] I get an error message saying that I should use Reduce for complete solution information, but I don’t understand how to do it.
I attach here the notebook that I use for this problem.