Example 1 Calculate $(x+y+z)(xy+yz+zx)$.
Analysis: Since both factors in the original expression are cyclic symmetric expressions in $x, y, z$, by property 5, their product is also a cyclic symmetric expression in $x, y, z$. Therefore, we only need to multiply the first letter of the first factor by the second factor, and then write down the remaining terms according to the cyclic symmetry rule.
Solution: Since $x(x y+y z+z x)=x^{2} y+x y z+z x^{2}$, therefore,
$$
\begin{array}{c}
\text { the original expression }=x^{2} y+x y z+z x^{2}+y^{2} z+y z x+x y^{2}+ \\
z^{2} x+z x y+y z^{2} \\
=x^{2} y+z x^{2}+y^{2} z+x y^{2}+z^{2} x+y z^{2}+3 x y z .
\end{array}
$$