Coordinate Geometry: JEE Main 2026 Mathematics question

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