We complete the square for both $x$ and $y$,
resulting in $(x+5)^2 - 25 + (y+12)^2 - 144 = 0$,
or $(x+5)^2 + (y+12)^2 = 169$.
This is the equation of a circle with center $(-5, -12)$ and radius 13.
Therefore, the area enclosed by this region is $\pi r^2 = \pi(13^2) = \boxed{169\pi}$.
The answer is: 169\pi
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
What is the area enclosed by the region defined by the equation $x^2+y^2+10x+24y=0$?

### Answer:
