Final answer:
Without additional context, it's not possible to select the correct polynomial for the measure of angle ABD. The options represent different potential relationships based on linear, quadratic, or cubic expressions.
Step-by-step explanation:
From the information provided, it's not possible to determine the exact polynomial that represents the measure of angle ABD without additional context regarding the geometric figure or the relationships between the angles. The polynomials provided as options (a) through (d) each represent different potential relationships. For instance, option (a) could represent a scenario where angle ABD is supplementary to another angle measuring x degrees, thus giving a measure of (180 - x) degrees. Option (b), (2x^2 - 5x + 90), could be used when angle ABD is part of a more complex geometric system where its size is related to a variable squared. Options (c) and (d), (x^3 - 3x^2 + 4x - 90) and (4x^2 - 6x + 180) respectively, would indicate an even more complex relationship which involves cubic terms or multiple terms of x squared.