To simplify this expression, we first need to find a common denominator for the fractions in the numerator:

2x/(x - 2) - 1/(x + 2) = (2x(x + 2) - 1(x - 2)) / ((x - 2)(x + 2))
= (2x^2 + 4x - x + 2) / (x^2 - 4)
= (2x^2 + 3x + 2) / (x^2 - 4)

Now, we can divide the resulting expression by the denominator in the second fraction:

(2x^2 + 3x + 2) / (x^2 - 4) ÷ (6x^2 + 9x + 6) / (x^2 - 4)

To divide two fractions, we can multiply by the reciprocal of the divisor:

(2x^2 + 3x + 2) / (x^2 - 4) * (x^2 - 4) / (6x^2 + 9x + 6)
= (2x^2 + 3x + 2) / (6x^2 + 9x + 6)

Therefore, the simplified expression is (2x^2 + 3x + 2) / (6x^2 + 9x + 6).