Problem

5. $f(x) = x^2 - 6x + 3$

The equation $f(x) = 0$ has roots $\alpha$ and $\beta$

Without solving the equation,

(a) determine the value of

$(\alpha^2 + 1)(\beta^2 + 1)$

(b) find a quadratic equation which has roots

$\dfrac{\alpha}{\alpha^2 + 1} \text{ and } \dfrac{\beta}{\beta^2 + 1}$

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