Undergrad Trouble Solving an Equation that has square roots on both sides

[thatboi]

Hey all,
I am having trouble solving the following equation for C
$$A(-\sqrt{C^2+4F_{+}}-C) = B(\sqrt{C^2+4F_{-}}+C)$$
I don't know how to get ride of the square roots on both sides.
Any help would be appreciated, thanks!

 

[fresh_42]

\begin{align*}
\sqrt{C^2+4F_{+}}&=D\sqrt{C^2+4F_{-}}+EC\\
C^2+4F_{+}&=D^2\left(C^2+4F_{-}\right)+E^2C^2+2DEC\sqrt{C^2+4F_{-}}\\
GC^2+H&=C\sqrt{C^2+4F_{-}}\\
G^2C^4+H^2+2GHC^2&=C^2(C^2+4F_{-})\\
\alpha C^4 + \beta C^2 +\gamma &=0
\end{align*}

Looks like some work to do because of the many constants, but doable.