At how many values of $x$ the following equality can hold

Question

I have the following equality $$x^{m/2}[m-x(m+2)]=x(1-x)(2b-1+x)(1-x-b)^{-2}$$ where $m>4$, $0<b<1$ and $0<x<1-b$. I know that the function on the right is a convex function while the function on the right is convex for $0<x<\frac{m^2-2m}{m^2+4m+4}$ and concave otherwise. I want to know at how many values of $x$ (where $0<x<1-b$) the above equality can hold. Thanks in advance.

Answer 1

note that for $m=6$ we get such polynomial in degree $5$ $$- \left( 8\,{x}^{5}+16\,b{x}^{4}-22\,{x}^{4}+8\,{b}^{2}{x}^{3}-28\,b{x }^{3}+20\,{x}^{3}-6\,{b}^{2}{x}^{2}+12\,b{x}^{2}-7\,{x}^{2}-2\,bx+2\,x +2\,b-1 \right) x =0$$