Completely simplify the following expression: $$ \frac{x^{2}-5 x+6}{x^{2}-9} $$ $$ \frac{x-1}{x+3} $$ $$ \frac{-5 x+6}{-9} $$ $$ \frac{x-6}{x-3} $$ $$ \frac{x-2}{x+3} $$

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To simplify the expression $$ \frac{x^{2}-5 x+6}{x^{2}-9} $$, we start by factoring both the numerator and the denominator.

1. **Factor the numerator $$ x^2 - 5x + 6 $$**:
   We need to find two numbers that multiply to $$ 6 $$ (the constant term) and add to $$ -5 $$ (the coefficient of $$ x $$). The numbers $$ -2 $$ and $$ -3 $$ work:
   $$
   x^2 - 5x + 6 = (x - 2)(x - 3)
   $$

2. **Factor the denominator $$ x^2 - 9 $$**:
   This is a difference of squares, which can be factored as:
   $$
   x^2 - 9 = (x - 3)(x + 3)
   $$

Now we can rewrite the original expression using these factorizations:
$$
\frac{x^2 - 5x + 6}{x^2 - 9} = \frac{(x - 2)(x - 3)}{(x - 3)(x + 3)}
$$

3. **Cancel the common factor $$ (x - 3) $$**:
   As long as $$ x \neq 3 $$ (to avoid division by zero), we can simplify:
   $$
   \frac{(x - 2) \cancel{(x - 3)}}{\cancel{(x - 3)}(x + 3)} = \frac{x - 2}{x + 3}
   $$

Thus, the completely simplified expression is:
$$
\boxed{\frac{x - 2}{x + 3}}
$$
The completely simplified expression is $$ \frac{x - 2}{x + 3} $$.

Completely simplify the following expression:

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