The given expression is:

$\frac{x^2 - x - 12}{3x - 9} \times \frac{x - 4}{12}$

Let's simplify it step by step.

Step 1: Factorize where possible

1. Factor the numerator $x^2 - x - 12$: $x^2 - x - 12 = (x - 4)(x + 3)$

2. Factor the denominator $3x - 9$: $3x - 9 = 3(x - 3)$

Now, substitute these factors back into the expression:

$\frac{(x - 4)(x + 3)}{3(x - 3)} \times \frac{x - 4}{12}$

Step 2: Combine the expressions

Now, multiply the two fractions:

$\frac{(x - 4)(x + 3)}{3(x - 3)} \times \frac{x - 4}{12} = \frac{(x - 4)(x + 3)(x - 4)}{3 \times 12 \times (x - 3)}$

Step 3: Simplify the expression

Multiply the constants in the denominator:

$3 \times 12 = 36$

So, the expression becomes:

$\frac{(x - 4)^2(x + 3)}{36(x - 3)}$

This is the simplified form of the expression.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you factor quadratic expressions in general?
2. What is the difference between simplifying and factoring?
3. How do you handle expressions with multiple fractions?
4. Can this expression be further simplified if given a specific value for $x$?
5. What are the common mistakes to avoid when simplifying algebraic fractions?

Tip: Always check for common factors in the numerator and denominator before multiplying fractions to simplify the process.