Find the equation for the ellipse that satisfies the given condition: 

Vertices $\left(\pm 5,0\right)$, foci $\left(\pm 4,0\right).$ 

It is given that vertices $\left(\pm 5,0\right)$ foci $\left(\pm 4,0\right)$ 
Clearly, the vertices are on the $x$-axis.
Therefore, the equation of the ellipse will be of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
where $a$ is the semi-major axis and $a=5,ae=4$