By Gauss's algorithm, if $\rm\ z^a\ y^b\ x^c\ $ is the highest w.r.t. lex order $\rm\ z > y > x\ $ then you subtract $\rm\ s_1^{a-b}\ s_2^{b-c}\ s_3^c\:.\:$ Thus since $\rm\ z^3\ y $ is highest you subtract $\rm s_1^{3-1}\ s_2^{1-0}\ s_3^0\ = (x+y+z)^2\ (xy+yz+zx)$ from $\rm\:P\:$. The result is smaller in lex-order, so iterating this reduction yields a representation of $\rm\:P\:$ in terms of elementary symmetric polynomials $\rm\:s_i\:.\:$ Here the algorithm terminates in two more steps.
As I mentioned in a prior post, Gauss's algorithm is the earliest known example of using lex-order reduction as in the Grobner basis algorithm. For a nice exposition see Chapter 7 of Cox, Little, O'Shea: Ideals, Varieties and Algorithms. They also give generalizations to the ring of invariants of a finite matrix group $\rm G \subset GL(n,k)$. Here's an excerpt which, coincidentally, presents this example. You might find it helpful to first read the example at the end before reading the proof.
