The problem is to simplify the expression $\frac{4a-12}{a+1} \div \frac{4}{a^2-1}$.

1. Problem Description

The problem is to simplify the expression $\frac{4a-12}{a+1} \div \frac{4}{a^2-1}$.

2. Solution Steps

First, we rewrite the division as multiplication by the reciprocal:

$$\frac{4a-12}{a+1} \div \frac{4}{a^2-1} = \frac{4a-12}{a+1} \cdot \frac{a^2-1}{4}$$

Next, we factor the expressions $4a-12$ and $a^2-1$:

$4a-12 = 4(a-3)$

$a^2-1 = (a-1)(a+1)$

Now, substitute the factored expressions into the equation:

$$\frac{4(a-3)}{a+1} \cdot \frac{(a-1)(a+1)}{4}$$

We can now cancel out the common factors of 4 and $(a+1)$:

$$\frac{4(a-3)}{a+1} \cdot \frac{(a-1)(a+1)}{4} = (a-3)(a-1)$$

Finally, expand the expression $(a-3)(a-1)$:

$$(a-3)(a-1) = a^2 - a - 3a + 3 = a^2 - 4a + 3$$

3. Final Answer

The simplified expression is $a^2 - 4a + 3$.