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.

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