2002 AMC 10B Problem 13

Grouping and factoring, $$\begin{aligned} &8xy - 12y + 2x - 3 \\ &= 4y(2x - 3) + (2x - 3) \\ &= (4y + 1)(2x - 3). \end{aligned}$$ For this to equal $0$ for all $y,$ the factor that depends on $y$ cannot be forced to zero, so we need $2x - 3 = 0,$ giving $x = \dfrac32.$ Thus, the correct answer is D.