4.2. 12 * Given that $x, y, z$ are positive numbers, and satisfy $x y z(x+y+z)=1$. Find the minimum value of the expression $(x+y)(x+z)$.

保留源文本的换行和格式，翻译结果如下：

4.2. 12 * Given that $x, y, z$ are positive numbers, and satisfy $x y z(x+y+z)=1$. Find the minimum value of the expression $(x+y)(x+z)$.
From the arithmetic mean being greater than or equal to the geometric mean, we get
$$
(x+y)(x+z)=y z+x(x+y+z) \geqslant 2 \sqrt{y z \cdot x(x+y+z)}-2,
$$

and when $x=\sqrt{2}-1, y=z=1$, the above inequality holds with equality. Therefore, the minimum value of the expression $(x+y)(x+z)$ is 2.