The problem is to simplify the expression $\frac{a-6}{a^2 - 9} - \frac{a+1}{a^2 + 2a - 3}$.

1. Problem Description

The problem is to simplify the expression $\frac{a-6}{a^2 - 9} - \frac{a+1}{a^2 + 2a - 3}$.

2. Solution Steps

First, we factor the denominators:

$a^2 - 9 = (a-3)(a+3)$

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

So the expression becomes:

$\frac{a-6}{(a-3)(a+3)} - \frac{a+1}{(a+3)(a-1)}$

The least common denominator is $(a-3)(a+3)(a-1)$.

Now we rewrite each fraction with the common denominator:

$\frac{(a-6)(a-1)}{(a-3)(a+3)(a-1)} - \frac{(a+1)(a-3)}{(a+3)(a-1)(a-3)}$

$\frac{a^2 - 7a + 6}{(a-3)(a+3)(a-1)} - \frac{a^2 - 2a - 3}{(a-3)(a+3)(a-1)}$

Now we combine the numerators:

$\frac{(a^2 - 7a + 6) - (a^2 - 2a - 3)}{(a-3)(a+3)(a-1)}$

$\frac{a^2 - 7a + 6 - a^2 + 2a + 3}{(a-3)(a+3)(a-1)}$

$\frac{-5a + 9}{(a-3)(a+3)(a-1)}$

So the simplified expression is $\frac{-5a+9}{(a-3)(a+3)(a-1)}$.

3. Final Answer

$\frac{-5a+9}{(a-3)(a+3)(a-1)}$