To simplify this expression, we need to find a common denominator for both fractions in the numerator and denominator.

4a² - 1 / a² - 9 can be factored as (2a + 1)(2a - 1) / (a + 3)(a - 3)

6a² + 3 / a + 3 can be factored as 3(2a + 1) / (a + 3)

Now, the expression becomes:

[(2a + 1)(2a - 1) / (a + 3)(a - 3)] / [3(2a + 1) / (a + 3)]

To simplify this further, we can multiply the numerator by the reciprocal of the denominator:

[(2a + 1)(2a - 1)(a + 3) / (a + 3)(a - 3)] * [(a + 3) / 3(2a + 1)]

Cancelling out common terms, we get:

(2a + 1)(2a - 1) / 3(a - 3)