It would help if you started by drawing a rough sketch of the region to be integrated. It should be clear that since it's bounded by x = 0, y = 0 and z = 0, that the region is only in the first octant. Then it will be bounded above by the paraboloid z = x^2 + y^2 + 6 (why would it be bounded above btw?) and bounded on the right by x + y = 2.
Can you come up with a triple integral from that?