Example 1. Solve the equation $\sqrt{x^{4}+2 x+5}$
$$
+\sqrt{x^{2}+2 x+37}=10
$$
$+\sqrt{(x+1)^{-}}+36=10$ , on the plane Cartesian coordinate system, the above equation indicates that the sum of the distances from point $P(x, 0)$ to points $F(-1, 2)$ and $F_{2}(-1,6)$ is 10. Therefore, $P$ lies on the ellipse $(a=5; c=2$, with the center at $(-1,4)$ )
$$
\frac{(x+1)^{2}}{21}+\frac{(y-4)^{2}}{25}=1
$$

. Setting $y=0$, we get $x=-1 \pm 3 \sqrt{21} / 5$.