1. **State the problem:** Find the equation of the ellipse with axes along the coordinate axes, vertices at $(\pm 5, 0)$, and foci at $(\pm 4, 0)$.\n\n2. **Recall the standard form of ellipse equation:** For an ellipse centered at the origin with major axis along the x-axis, the equation is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$$ where $a$ is the semi-major axis length and $b$ is the semi-minor axis length.\n\n3. **Identify given values:** Vertices at $(\pm 5, 0)$ imply $a = 5$. Foci at $(\pm 4, 0)$ imply $c = 4$, where $c$ is the focal distance from the center.\n\n4. **Use the relationship between $a$, $b$, and $c$ for ellipses:** $$c^2 = a^2 - b^2.$$\n\n5. **Calculate $b^2$:** $$b^2 = a^2 - c^2 = 5^2 - 4^2 = 25 - 16 = 9.$$\n\n6. **Write the equation of the ellipse:** $$\frac{x^2}{25} + \frac{y^2}{9} = 1.$$\n\n**Final answer:** The equation of the ellipse is $$\frac{x^2}{25} + \frac{y^2}{9} = 1.$$