Example 2. Solve the inequality
$$
\sqrt{x^{2}+6 x+10}+\sqrt{x^{2}-6 x+10}>10 .
$$
Solve the inequality which can be transformed into
$$
\sqrt{(x+3)^{2}+1}+\sqrt{(x-3)^{2}+1}>10 \text {. }
$$

Let $1=y^{2}$, then we have
$$
\sqrt{(x+3)^{2}+y^{2}}+\sqrt{(x-3)^{2}+y^{2}}>10 \text {. }
$$

This precisely indicates that the sum of the distances from the moving point $(x, y)$ to the points $(3,0)$ and $(-3, 0)$ is greater than 10. That is, $(x, y)$ lies outside the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$. Therefore, the inequality is equivalent to $\frac{x^{2}}{25}+\frac{y^{2}}{16}>1$ $\left(y^{2}=1\right)$. Solving this, we get
$$
x>\frac{5}{4} \sqrt{15} \text { or } x<-\frac{5}{4} \sqrt{15} \text {. }
$$