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