Newton's Sums

Algebra Level 3

If the roots of \(p(x) = x^3 + 3x^2 + 4x - 8\) are \(\color{red}{a}\), \(\color{blue}{b}\) and \(\color{purple}{c}\), what is the value of

\[\color{red}{a}^2 \big(1 + \color{red}{a}^2\big) + \color{blue}{b}^2 \big(1 + \color{blue}{b}^2\big) + \color{purple}{c}^2 \big(1 + \color{purple}{c}^2\big)?\]