Problem

The equation $2x^2 + 5x + 7 = 0$ has roots $\alpha$ and $\beta$

Without solving the equation

(a) determine the exact value of $\alpha^3 + \beta^3$

(b) form a quadratic equation, with integer coefficients, which has roots

$\frac{\alpha^2}{\beta} \quad \text{and} \quad \frac{\beta^2}{\alpha}$