We are asked to find the quotient of the expression:

$\frac{-4x - 12}{x - 7} \div \frac{x^2 - 12x + 36}{2x - 14}$

Step 1: Rewrite the division as multiplication by the reciprocal

We know that dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite the expression as:

$\frac{-4x - 12}{x - 7} \times \frac{2x - 14}{x^2 - 12x + 36}$

Step 2: Factor the expressions

Let's factor each part of the expression where possible:

1. Factor $-4x - 12$: $-4x - 12 = -4(x + 3)$

2. Factor $x - 7$: This is already in its simplest form.

3. Factor $2x - 14$: $2x - 14 = 2(x - 7)$

4. Factor $x^2 - 12x + 36$ (this is a perfect square trinomial): $x^2 - 12x + 36 = (x - 6)^2$

Step 3: Substitute the factored forms back into the expression

Now that we have factored each part, substitute them back into the original equation:

$\frac{-4(x + 3)}{x - 7} \times \frac{2(x - 7)}{(x - 6)^2}$

Step 4: Simplify by canceling common factors

Notice that $x - 7$ appears in both the numerator and denominator, so we can cancel it out:

$\frac{-4(x + 3)}{1} \times \frac{2}{(x - 6)^2}$

Now, multiply the numerators and denominators:

$\frac{-4(x + 3) \times 2}{(x - 6)^2}$

This simplifies to:

$\frac{-8(x + 3)}{(x - 6)^2}$

Step 5: Final simplified quotient

The quotient in its lowest terms is:

$\boxed{\frac{-8(x + 3)}{(x - 6)^2}}$

This is the final simplified expression.

Do you want any further clarification or details?

Here are 5 questions that might expand your understanding:

1. How do you factor a trinomial like $x^2 - 12x + 36$?
2. What are the steps to simplify rational expressions?
3. Why does dividing by a fraction turn into multiplying by the reciprocal?
4. How do you know when you can cancel terms in a fraction?
5. What are common types of factoring methods used in algebra?

Tip: When simplifying rational expressions, always look for common factors in the numerator and denominator to cancel out before multiplying.