jmc

The radius of the larger circle is given by the distance formula to be $\sqrt{(6-(-1){)}^2+(1-0{)}^2}=\sqrt{49+1}=5\sqrt{2}$. The distance between the centers of the two circles is given by the distance formula to be $\sqrt{(-3-0{)}^2+(2-(-1){)}^2}=\sqrt{3^2+3^2}=3\sqrt{2}$. Thus, the radius of the smaller circle is equal to $5\sqrt{2}-3\sqrt{2}=2\sqrt{2}$, and the square of the radius is $8$. The equation of the smaller circle is given by $$(x+3{)}^2+(y-2{)}^2=8⟹x^2+y^2+6x-4y+5=0.$$ So $D+E+F=6-4+5=\boxed{7}$.