Limit, Continuity and Differentiability: JEE Main 2024 Mathematics question

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $(1+x(\lambda^2-x^2))$ satisfies $\frac{x^2+x+2}{x^2+5 x+6}<0$, be $(\alpha, \beta)$. Then $\alpha^2+\beta^2$ is equal to _________.