The equation provided is:

$\frac{x^2 - 5x + 6}{xy + x - 3y - 3}$

To simplify this expression, let's start by factoring both the numerator and the denominator.

1. Factoring the numerator:

The numerator is $x^2 - 5x + 6$. This quadratic can be factored as:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

2. Factoring the denominator:

The denominator is $xy + x - 3y - 3$. We can factor this expression by grouping:

$xy + x - 3y - 3 = x(y + 1) - 3(y + 1)$

Now, factor out the common factor $(y + 1)$:

$xy + x - 3y - 3 = (x - 3)(y + 1)$

3. Simplifying the expression:

The original expression simplifies to:

$\frac{(x - 2)(x - 3)}{(x - 3)(y + 1)}$

Notice that $(x - 3)$ appears in both the numerator and the denominator, so they cancel each other out (assuming $x \neq 3$):

$\frac{x - 2}{y + 1}$

So, the simplified expression is:

$\frac{x - 2}{y + 1}$

Would you like any additional details or have any questions?

Here are 5 related questions that expand on this information:

1. How do you factor more complex quadratic expressions?
2. What happens if the expression involves variables that cannot be factored easily?
3. How can you verify if your simplification is correct?
4. What are the conditions under which certain terms can be canceled in a fraction?
5. How would the process change if the denominator was more complicated?

Tip: Always check the domain of the variables when canceling terms to ensure you're not dividing by zero.