This depends on the precise definitions of angles, complementary angles and supplementary angles. There are at least three variants.

(1) An angle is just a number. Then, of course, there exist solutions to the equations -42 + x = 90 and -42 + x = 180.

(2) An angle is a geometric figure on a plane consisting of two rays with a common beginning. Then it does not make sense to talk about angles whose measure is negative.

(3) This is like (2), but we fix a direction of the positive rotation on the plane (say, counterclockwise) and designate which of the angle's sides is the first and which is the second. Then the angle's measure is considered positive if the first side, rotated by that measure in the positive direction, coincides with the second side. Geometric addition of angles should be made more specific by saying that we make the second side of the first angle coincide with the first side of the second angle.

In (3), every angle has a complementary and a supplementary angle. For example, a complementary angle to -42 is 132.

I think -42 lies in the fourth quadrant.I do not think that it can happen. my angle is in the third quadrant.