10. Solve the equation $\sqrt{x^{2}+6 x+13}+\sqrt{x^{2}-2 x+4}=8$.
10. The original equation is equivalent to $\left\{\begin{array}{l}\sqrt{(x+3)^{2}+y^{2}}+\sqrt{(x-1)^{2}+y^{2}}=8, \\ y^{2}=3 .\end{array}\right.$

Construct an ellipse: with center at $(-1,0)$, foci at $F_{1}(-3,0)$ and $F_{2}(1,0)$, and $2 a=8$, yielding the ellipse equation $\frac{(x+1)^{2}}{16}+\frac{y^{2}}{12}=1$. Let $y^{2}=3$, then $x_{1,2}=-1 \pm 2 \sqrt{3}$ are the solutions.