I would do this in two parts.2) Consider that part of the parabola y = (x-1)*(x+2) which lies below the x-axis. Calculate the volume of the solid formed when this part is rotated about the line y =4.
First look at the parabola only. Imagine a line from to y= 4. Rotating around the axis y= 4 gives a disk of radius and so of area . Taking each disk to have "thickness" dx, The volume of each disk is and the volume of all put together is . The integral is, of course, from x= -2 to x= 1 where the parabola crosses the x-axis.
Second, subtract the volume of the cylinder made up of the portion that is above the x-axis. That will be a cylinder with radius 4 and height 1-(-2)= 3.