Problem

3. The quadratic equation

$5x^2 + 2x + 3 = 0$

has roots $\alpha$ and $\beta$

Without solving the equation,

(a) write down the value of $(\alpha + \beta)$ and the value of $\alpha\beta$

(b) determine the value of

(i) $\alpha^2 + \beta^2$

(ii) $\alpha^3 + \beta^3$

(c) determine a quadratic equation that has roots

$(\alpha + \beta^2) \text{ and } (\beta + \alpha^2)$

giving your answer in the form $px^2 + qx + r = 0$ where $p$, $q$ and $r$ are integers.