4a^2-1/a^2-9

To simplify the expression \( \frac{4a^2 - 1}{a^2 - 9} \), we can start by factoring both the numerator and the denominator.

1. Factor the numerator: The numerator \( 4a^2 - 1 \) is a difference of squares and can be factored as: \[ 4a^2 - 1 = (2a - 1)(2a + 1) \]

2. Factor the denominator: The denominator \( a^2 - 9 \) is also a difference of squares and can be factored as: \[ a^2 - 9 = (a - 3)(a + 3) \]

3. Rewrite the expression: Now we can rewrite the original expression with the factored forms: \[ \frac{4a^2 - 1}{a^2 - 9} = \frac{(2a - 1)(2a + 1)}{(a - 3)(a + 3)} \]

Since there are no common factors in the numerator and the denominator, this is the simplified form of the expression.

Thus, the simplified expression is: \[ \frac{(2a - 1)(2a + 1)}{(a - 3)(a + 3)} \]