Example 1. Solve the equation
$$
\sqrt{x^{2}+4 x+8}+\sqrt{x^{2}-8 x+20}=10 \text {. }
$$
Analysis: Removing the square root from this equation is not easy, but if we let $y^{2}=4$, the original equation becomes $\sqrt{(x+2)^{2}+y^{2}}+\sqrt{(x-4)^{2}+y^{2}}=10$, from which we know that the trajectory of $(x, y)$ is an ellipse $\frac{(x-1)^{2}}{25}+\frac{y^{2}}{16}=1$. Substituting $y^{2}=4$ into it, we find that $x=1 \pm \frac{5 \sqrt{3}}{2}$.