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.

$P(s,t)=s^4+t^4-2s^2-2t^2+2$. 
Now, $s^2+t^2=\sigma^2-2\pi$, where $\sigma$ and $\pi$ are the sum and product of $s$ and $t$. Thus, $s^4+t^4=(s^2+t^2)^2-2s^2t^2=(\sigma^2-2\pi)^2-2\pi^2=\sigma^4-4\pi\sigma^2+2\pi^2$. 
So $P(s,t)=2-2\sigma^2+4\pi+\sigma^4-4\pi\sigma^2+2\pi^2$. 
