jmc

We complete the square on the quadratic in $x$ by adding $(10/2{)}^2=25$ to both sides, and complete the square on the quadratic in $y$ by adding $(24/2{)}^2=144$ to both sides. We have the equation $$(x^2+10x+25)+(y^2+24y+144)=169\Rightarrow (x+5{)}^2+(y+12{)}^2=169$$We see that this is the equation of a circle with center $(-5,-12)$ and radius 13. Thus, the area of the region enclosed by this circle is $\pi \cdot 13^2=\boxed{169\pi }$.