Originally Posted by
red_dog Graphic.
We have $\displaystyle m=-\frac{1}{9}x^2-\frac{2}{3}x$
Intersect the graph of $\displaystyle f(x)=-\frac{1}{9}x^2-\frac{2}{3}x$ with parallel lines to x-axis with equation $\displaystyle y=m$
If $\displaystyle m>1$ then the line doesn't intersect the graph.
If $\displaystyle m=1$ the line is tangent to the graph and the equation has a double solution $\displaystyle x=-3$
If $\displaystyle m\in(0,1)$ the line intersects the graph in 2 points and the equation has two negative solutions.
If $\displaystyle m=0$ then the equation has the solutions $\displaystyle x_1=-6, \ x_2=0$
If $\displaystyle m<0$ then the equation has one negative solution and one positive solution.