Newton's sum and complex numbers

Question

Does Newton's sums include imaginary numbers? For example, if $α,β$ are the real roots of $x^4-2x^2-1=0,$ and I want to calculate $\frac{(α^2+β^2)(α^8+β^8)}{α^4+β^4}$, can I use Newton's sums?

Answer 1

Yes, it does.
However for this particular problem you can complete the square to obtain $$x^2=1\pm\sqrt2.$$ Then through away the complex roots :) to obtain the values of $\alpha^2$ and $\beta^2.$