The thing to notice is that the quadratic formula tells you what you're two (possibly repeated) roots are going to be. For real roots, which you need to have $\alpha<a<\beta$, the discriminant must be positive. Moreover, the least root \alpha will be of the form $c-d\sqrt\Delta$, where $\Delta$ is your discriminant, and the greater root $\beta$ will be of the form $c+d\sqrt\Delta$. You then have a system of two inequalities, $\alpha<a$ and $\beta>a$.
In particular, you need to solve the following two inequalities:
There is one more implicit inequality for the solutions to make sense. Since your answers must be distinct reals, you need your discriminant positive $$4(2a^2+6a+1)>0.$$