For what values of $a$ is the sum of the roots of the equation $x^2 + \left(2-a-a^2\right)x - a^2 = 0$ equal to zero?
Given quadratic equation: $\;\;\;$ $x^2 + \left(2-a-a^2\right)x - a^2 = 0$ $\;\;\; \cdots \; (1)$
Let the roots of $(1)$ be $\alpha$ and $\beta$
Sum of roots $= \alpha + \beta = \dfrac{-\left(2 - a - a^2\right)}{1} = a^2 + a - 2$
as per question, $\;$ $\alpha + \beta = 0$
i.e. $\;$ $a^2 + a - 2 = 0$
i.e. $\;$ $\left(a + 2\right) \left(a - 1\right)= 0$
i.e. $\;$ $a = -2$ $\;$ or $\;$ $a = 1$