$$ f(x)=\frac{\sqrt{x^2-25}}{4-x^2}+\log _{10}\left(x^2+2 x-15\right) $$ Domain : $\mathrm{x}^2-25 \geq 0 \Rightarrow \mathrm{x} \in(-\infty,-5] \cup[5, \infty)$ $$ \begin{aligned} & 4-x^2 \neq 0 \Rightarrow x \neq\{-2,2\} \\ & x^2+2 x-15>0 \Rightarrow(x+5)(x-3)>0 \\ & \Rightarrow x \in(-\infty,-5) \cup(3, \infty) \\ & \therefore x \in(-\infty,-5) \cup[5, \infty) \end{aligned} $$ $$ \begin{aligned} & \alpha=-5 ; \beta=5 \\ & \therefore \alpha^2+\beta^3=150 \end{aligned} $$