To determine the common ratio $r$ of the geometric sequence with terms $x+2$, $2x$, and $4x-6$, we can proceed as follows:

Solution Steps:

1. Identify the property of a geometric sequence:
   For any three consecutive terms $a$, $ar$, and $ar^2$ in a geometric sequence, the ratio of consecutive terms is constant. Thus: $\frac{2x}{x+2} = \frac{4x - 6}{2x}$

2. Set up the first ratio equation: $\frac{2x}{x+2} = \frac{4x - 6}{2x}$

3. Cross-multiply to eliminate the fractions: $2x \cdot 2x = (x + 2)(4x - 6)$ [ 4x^2 = 4x^2 - 6x + 8x -12