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If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 + x - 2$, then find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.
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Here $\alpha + \beta = - 1$ and $$\begin{aligned}& \alpha\beta = -2 \\ & \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta} \\ & = \frac{(-1)^2-2(-2)}{-2} = \frac{1+4}{-2} = -\frac{5}{2}\end{aligned}$$