Example 5 Find the prime solutions of the equation $x(x+y)=z+120$.

保持源文本的换行和格式，翻译结果如下：

Example 5 Find the prime solutions of the equation $x(x+y)=z+120$.
Solution: (1) When $z$ is even, since $z$ is a prime number, we get $z=2$. At this time, the equation transforms into: $x(x+y)=122$. Since $x$ is a prime number, $x=2$, thus we get $y=59$, and 59 is a prime number. Therefore, when $z$ is even, the prime number solution of the original equation is $(2,59,2)$.
(2) When $z$ is odd, $x$ and $(x+y)$ must be odd, which implies that $y$ should be even. Since $y$ is a prime number, then $y=2$. Thus, the equation transforms into $(x-10)(x+12)=z$. Since $z$ is an odd prime number, one of $x-10$ and $x+12$ must be 1, so we have $x-10=1$, i.e., $x=11$. Therefore, $z=23$.
Hence, when $z$ is odd, the prime number solution of the equation is $(11,2,23)$.
In summary, the prime number solutions of the original equation are $(2,59,2)$ and $(11,2,23)$.