jmc

We rewrite the equation as $x^2+14x+y^2-4y=10$ and then complete the square, resulting in $(x+7{)}^2-49+(y-2{)}^2-4=10$, or $(x+7{)}^2+(y-2{)}^2=63$. This is the equation of a circle with center $(-7,2)$ and radius $\sqrt{63},$ so the area of this region is $\pi r^2=\boxed{63\pi }$.