Solution review
I think the gadget or the hand-drawing is too much. I think that despite the fact that the function is fairly easy (being a polynomial for which we see all the roots) it is more likely that students will run all the regular tools they have, and so we could provide sign tables (for the function, its derivative and second derivative).
As for the hint, we could ask them something a little more conceptual at first, maybe about how do they find information about the graph of a function, or more specifically, how do they know about intervals of increase, decrease, and concavity and such?
The official exam solution for this question gave points for identifying the roots of the equation, the local max and min (they didn't have to compute derivatives and solve for critical points, but had to mark them on the graph), and for showing that the functions increased / decreased without bound appropriately as |x| gets bigger. At that point, as long as they connected the dots they were allotted full marks.
This was meant just as a visual aid for part b of this question, so that they could pick out the correct bounds for integration.
So the marking scheme reflects the way mathematicians would think of this problem. I still think we should focus on what will help students improve their skills, whatever that might be. I'm not attached to a particular way of solving this question, I just wonder what best reflects our commitment to provide a useful resource to students.
I have attempted this problem considering the recommendations that are mentioned here. I plotted the graph highlighting all the special attributes in a different way. Do we think this is sufficient now?