jmc

The quadratic formula yields $$x=\frac{-(a+8)\pm \sqrt{(a+8{)}^2-4\cdot 4\cdot 9}}{2\cdot 4}.$$The equation has only one solution precisely when the value of the discriminant, $(a+8{)}^2-144$, is 0. This implies that $a=-20$ or $a=4$, and the sum is $\boxed{-16}$.