Answer
$$
\left(\frac{x}{10}\right)^{2}+\left(\frac{y}{8}\right)^{2}=1.
$$
Work Step by Step
Since the vertices are at $(\pm10,0)$ and the foci are at $(\pm6, 0)$, then the foci are at the x-axis
$$a=10, \quad b^2=a^2-c^2=100-36=64$$
and hence the ellipse is centered at the origin and has the equation
$$
\left(\frac{x}{10}\right)^{2}+\left(\frac{y}{8}\right)^{2}=1.
$$
