The given expression is:

$\frac{2x^2 - 4x - 16}{x^2 - x - 12} \div \frac{x - 2}{x^2 + 7x + 12}$

Step 1: Simplify each part of the expression

Numerator (first fraction):

$2x^2 - 4x - 16$ We can factor out $2$: $2(x^2 - 2x - 8)$ Now, factor $x^2 - 2x - 8$: $x^2 - 2x - 8 = (x - 4)(x + 2)$ Thus, the numerator becomes: $2(x - 4)(x + 2)$

Denominator (first fraction):

$x^2 - x - 12$ We need to factor this quadratic. Look for two numbers that multiply to $-12$ and add to $-1$: $x^2 - x - 12 = (x - 4)(x + 3)$

So, the first fraction becomes: $\frac{2(x - 4)(x + 2)}{(x - 4)(x + 3)}$

Second fraction:

$\frac{x - 2}{x^2 + 7x + 12}$ Factor $x^2 + 7x + 12$: $x^2 + 7x + 12 = (x + 3)(x + 4)$

So the second fraction is: $\frac{x - 2}{(x + 3)(x + 4)}$

Step 2: Rewrite the division as multiplication

To divide by a fraction, we multiply by its reciprocal: $\frac{2(x - 4)(x + 2)}{(x - 4)(x + 3)} \times \frac{(x + 3)(x + 4)}{x - 2}$

Step 3: Simplify the product

Now cancel out any common factors. Notice that $(x - 4)$ and $(x + 3)$ appear in both the numerator and the denominator:

$= \frac{2(x + 2)(x + 4)}{x - 2}$

This is the simplified expression.

Step 4: Final Answer

The simplified expression after performing the division is:

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

Would you like further details on any of the steps?

Questions to expand:

1. What are the properties of multiplying and dividing rational expressions?
2. How do you factor quadratics like $x^2 - x - 12$?
3. How can you simplify complex rational expressions?
4. Why is it important to check for common factors before multiplying fractions?
5. How do you handle undefined values for rational expressions (e.g., what happens if $x = 2$)?

Tip:

Always factor expressions completely before performing operations like multiplication or division to ensure the simplest form.