Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$? Consider the symmetric polynomial
$$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$
How can we write $P$ as a polynomial in the variables $st,t+s$?
The Fundamental theorem of symmetric polynomials implies this is possible, but I am having trouble doing it in practice.

\begin{align}
& P(s, t) = (s^2 - 1 + t^2 - 1)^2 - 2(s^2 - 1)(t^2 - 1) \\
= & ((s + t)^2 - 2st - 2)^2 - 2[s^2t^2 - (s^2 + t^2) + 1] \\
= & ((s + t)^2 - 2st - 2)^2 - 2[s^2t^2 - (s + t)^2 + 2st + 1]
\end{align}
