problem: Consider the circle defined by the equation $x^2 +6x +y^2 +8y =0$.  Find the sum of the coordinates of the center of the circle.
solution: We complete the square for the equation by observing that the given equation is equivalent to \[
  (x^2+6x+9)+(y^2+8y+16)=25,
  \] so the equation for the circle becomes \[
  (x+3)^2 +(y+4)^2 =5^2.
  \]  Hence, the center is $(-3,-4)$ which implies that the answer is $-3-4 =\boxed{-7}$.