To simplify the expression $\frac{x^3 - 3x^2 - x + 3}{x^2 - 5x + 6}$, we can start by factoring both the numerator and the denominator.

Step 1: Factor the denominator

The denominator is $x^2 - 5x + 6$. We need to find two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. So, the denominator factors as: $x^2 - 5x + 6 = (x - 2)(x - 3)$

Step 2: Factor the numerator

Next, we factor the numerator $x^3 - 3x^2 - x + 3$. We can try factoring by grouping: $x^3 - 3x^2 - x + 3 = (x^3 - 3x^2) - (x - 3)$ Factor out $x^2$ from the first group and $-1$ from the second group: $= x^2(x - 3) - 1(x - 3)$ Now, factor out the common factor $(x - 3)$: $= (x - 3)(x^2 - 1)$ Notice that $x^2 - 1$ is a difference of squares and can be factored further: $x^2 - 1 = (x - 1)(x + 1)$ So, the numerator fully factors as: $x^3 - 3x^2 - x + 3 = (x - 3)(x - 1)(x + 1)$

Step 3: Simplify the expression

Now, the expression is: $\frac{(x - 3)(x - 1)(x + 1)}{(x - 2)(x - 3)}$ We can cancel out the common factor $(x - 3)$ from the numerator and the denominator: $= \frac{(x - 1)(x + 1)}{x - 2}$ Thus, the simplified expression is: $\boxed{\frac{(x - 1)(x + 1)}{x - 2}}$

Would you like any further details or have any questions about this problem?

Here are 5 related questions you might be interested in:

1. How do you solve rational expressions with different factors in the numerator and denominator?
2. What is the importance of factoring in algebra?
3. Can you explain how to solve equations involving rational expressions?
4. How do you handle undefined values in rational expressions?
5. What are some real-world applications of simplifying rational expressions?

Tip: Always check for common factors between the numerator and the denominator before simplifying an expression to avoid unnecessary steps.