Finding the missing number in an exponential polynomial series

Extract the mathematical blueprint of the sequence using finite differences: Sequence: 2, 12, 36, 80, 150 First differences (D_1): 10, 24, 44, 70 Second differences (D_2): 14, 20, 26 Third differences (D_3): 6, 6

The third-order difference is a constant 6, marking this as a cubic polynomial sequence. Project the pattern forward: Next D_2 = 26 + 6 = 32. Next D_1 = 70 + 32 = 102. Next sequence term X = 150 + 102 = 252.

(Alternate analytical bypass): Check against standard cubic templates: 1³ + 1² = 1 + 1 = 2 2³ + 2² = 8 + 4 = 12 3³ + 3² = 27 + 9 = 36 4³ + 4² = 64 + 16 = 80 5³ + 5² = 125 + 25 = 150 The 6th term matches: 6³ + 6² = 216 + 36 = 252.

Answer: (b).