A family of functions $\{f_{a,b} \mid a, b \in \mathbb{R}\}$ is given thru $f_{a,b}(x)=\frac{x^4+ax^2+b}{x^2+1}.$ It is asked to pick those functions such that their graph $\Gamma_{a,b}$ is tangent to the horizontal axis in two distinct points.
I have written the conditions of tangency and of passage thru a point $(x,0)$. The discussion is a bit long, but possible. I wonder if there is any "shortcut", i.e. some approach which leads to the solution more quickly.