I'm not really sure what you're asking? You would integrate if you wanted to find the area under the curve, and the question seems to be asking you to find the equation of the curve.
One of the solutions to y=0 is x=0 so the quadratic would have the form y=a(x^2)+bx which is the same as y=x(ax+b). Next sub in (-2,3) to get 3=4a-2b. Now you need to differentiate y=a(x^2)+bx which gives dy/dx=2ax+b. At x=-2 dy/dx is -1 so we have -4a+b=-1. Then you have two simultaneous equations that you can solve to get a and b, no integration required.
One of the solutions to y=0 is x=0 so the quadratic would have the form y=a(x^2)+bx which is the same as y=x(ax+b). Next sub in (-2,3) to get 3=4a-2b. Now you need to differentiate y=a(x^2)+bx which gives dy/dx=2ax+b. At x=-2 dy/dx is -1 so we have -4a+b=-1. Then you have two simultaneous equations that you can solve to get a and b, no integration required.
You're not supposed to just give the answer. Its a student forum, not an answer forum.