2 The quadratic equation
$$2 x ^ { 2 } + 8 x + 1 = 0$$
has roots \(\alpha\) and \(\beta\).
- Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
- Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
- Hence, or otherwise, show that \(\alpha ^ { 4 } + \beta ^ { 4 } = \frac { 449 } { 2 }\).
- Find a quadratic equation, with integer coefficients, which has roots
$$2 \alpha ^ { 4 } + \frac { 1 } { \beta ^ { 2 } } \text { and } 2 \beta ^ { 4 } + \frac { 1 } { \alpha ^ { 2 } }$$
\includegraphics[max width=\textwidth, alt={}]{2eaee88a-9e08-4392-8a4c-79fc9861603e-04_2443_1707_260_153}