To simplify the expression (9x^2 - 1)/(x^2 - 25) ÷ (3x - 1)/(6x + 30), we first factor the numerator and denominator of the first fraction:

9x^2 - 1 = (3x + 1)(3x - 1)
x^2 - 25 = (x + 5)(x - 5)

Therefore, the expression becomes:

((3x + 1)(3x - 1))/((x + 5)(x - 5)) ÷ (3x - 1)/(6(x + 5))

Next, we simplify it further by multiplying the fractions:

((3x + 1)(3x - 1))/((x + 5)(x - 5)) * (6(x + 5))/(3x - 1)

Now, we can cancel out the common factors in the numerator and denominator:

(3x + 1) 2(x + 5) = 2(3x + 1)(x + 5)
(x + 5) 2(3x + 1) = 2(3x + 1)(x + 5)

Therefore, the simplified expression is:

2(3x + 1)(x + 5) / (3x - 1)