Ellipse

Let $\frac{x^2}{f\left(a^2+7 a+3\right)}+\frac{y^2}{f(3 a+15)}=1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function on $\mathbf{R}$. If the set of all possible values of $a$ is $\mathbf{R}-[\alpha, \beta]$, then $\alpha^2+\beta^2$ is equal to :