Here $c_1,c_2,c_3$ are constants bounded in absolute value by one, and at least one of them is nonzero. It seems "obvious" from looking at graphs that the answer is at most three. Rolle's Theorem leads nowhere for me since the derivative is more complicated than the function itself. By changing variables $t\to -\log u$ and manipulating things a bit, I can show that the roots occur at intersections of the functions $\log^2u$ and $$\frac{{c_2} (u-1)^2 u}{{c_1} (u-1)^2-4 {c_3} u^2}$$ for $u\in [0,1]$. From here it is tempting to invoke a false theorem that monotone convex functions have at most two intersections.
The only other option I can think of is to break the parameter space into pieces and make clever use of inequalities on each portion of the space. This seems like a lot of work, so before going that route I wondered if there are any other tricks I am missing.