you have:

just graph the parabola it will be easy to identify the interval where it is less than or equal to zero (on this interval, the graph will be below the x-axis). so the answer is the range of x's for which the graph is under the x-axis (this includes the end points, since we can be equal to zero)

this will indeed be easier if you consider the factors (x + 3)(x - 4). this is zero for x = -3, and x = 4. now check all regions in between those and on either side to see if the inequality holds (your professor was trying to emphasize the concept that we have a number less than zero if we have a positive times a negative, so just make sure one is negative while the other is positive)