To find the area of the quadrilateral, we need to find the area of the trapezoid and the area of the right triangle at the bottom separately, and then add them together.
Area of trapezoid = (1/2) * (sum of parallel sides) * (height)
Area of trapezoid = (1/2) * (4 + 6) * 9
Area of trapezoid = (1/2) * 10 * 9
Area of trapezoid = 45 cm²
Area of right triangle = (1/2) * base * height
Area of right triangle = (1/2) * 4 * 3
Area of right triangle = 6 cm²
Total area = Area of trapezoid + Area of right triangle
Total area = 45 cm² + 6 cm²
Total area = 51 cm²
Therefore, the correct answer is 51 cm².