Unclassified (maths): JEE Main 2025 Mathematics question

Let the function, $f(x) = \begin{cases} -3 a x^2 - 2, & x < 1 \\ a^2 + b x, & x \geqslant 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1$, $b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is: