In a quadrilateral, how do I find the measure of angle AHM that is 3x+5 and another angle TMH is 7x-27?

You don't have enough information to say what any of the angles are. The quadrilateral has four internal angles that add up to 360 degrees, and you only know two of them in terms of x.

If it were a parallelogram, there would be two pairs of equal angles, and you could then solve for x and all angles