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.