jmc

We solve the equation $f(x)=-5$ on the domains $x<-2$ and $x\ge -2.$

If $x<-2,$ then $f(x)=2x+7,$ so we want to solve $2x+7=-5.$ The solution is $x=-6,$ which satisfies $x<-2.$

If $x\ge -2,$ then $f(x)=-x^2-x+1,$ so we want to solve $-x^2-x+1=-5.$ This equation simplifies to $x^2+x-6=0,$ which factors as $(x-2)(x+3)=0.$ The solutions are $x=2$ and $x=-3,$ but only $x=2$ satisfies $x\ge -2.$

Therefore, the solutions are $-6$ and $2,$ and their sum is $(-6)+2=\boxed{-4}.$