$F(x)=(x^2+ax+b)e^{-x}\\ F'(x)=(2x+a)e^{-x}-(x^2+ax+b)e^{-x}\\ =(-x^2+(2-a)x+a-b)e^{-x}\\ =f(x)=(-x^2+3x+6)e^{-x}\\ \Rightarrow \left\{\begin{array}{l} 2-a=3\\a-b=6\end{array} \right.\\ \Rightarrow \left\{\begin{array}{l} a=-1\\b=-7\end{array} \right.$