We are tasked with simplifying the given expression and identifying the excluded values of $x$ that make any denominator equal to zero. Let's break it down.

Expression:

$\frac{-4x - 32}{x^2 - 12x + 36} \times \frac{x^2 - 9x + 18}{3x + 24}$

Step 1: Factor each part of the expression.

Factor the denominators and numerators.

1. Factor $x^2 - 12x + 36$:

This is a quadratic trinomial that factors as:

$x^2 - 12x + 36 = (x - 6)^2$

2. Factor $3x + 24$:

This is a linear expression that can be factored by taking out the greatest common factor (GCF):

$3x + 24 = 3(x + 8)$

3. Factor $x^2 - 9x + 18$:

This is a quadratic trinomial that factors as:

$x^2 - 9x + 18 = (x - 6)(x - 3)$

4. Factor $-4x - 32$:

We can factor out $-4$ from this expression:

$-4x - 32 = -4(x + 8)$

Step 2: Rewrite the expression with the factors:

$\frac{-4(x + 8)}{(x - 6)^2} \times \frac{(x - 6)(x - 3)}{3(x + 8)}$

Step 3: Simplify the expression.

Now we can cancel common factors:

• $x + 8$ in the numerator and denominator cancels out.
• $x - 6$ in the numerator and denominator cancels out one of the $(x - 6)$'s in the denominator.

The simplified expression becomes:

$\frac{-4(x - 3)}{3(x - 6)}$

Step 4: Identify excluded values.

We exclude any $x$-values that make the original denominators equal to zero.

1. From $x^2 - 12x + 36 = (x - 6)^2$:
   $x - 6 = 0$ implies $x = 6$.

2. From $3x + 24 = 3(x + 8)$:
   $x + 8 = 0$ implies $x = -8$.

Thus, the values that must be excluded are $x = 6$ and $x = -8$.

Final Answer:

• The product in lowest terms is: $\frac{-4(x - 3)}{3(x - 6)}$.
• The values of $x$ to exclude are: $x = 6$ and $x = -8$.

Choices:

• (Choice A) $x = -8$ — Exclude.
• (Choice B) $x = 0$ — Not excluded.
• (Choice C) $x = 3$ — Not excluded.
• (Choice D) $x = 6$ — Exclude.

Thus, the correct choices are A and D.

Would you like more details or have any questions?
Here are 5 related questions to consider:

1. How do you factor a quadratic trinomial like $x^2 - 12x + 36$?
2. What are the steps for simplifying complex fractions like this?
3. Why do we exclude certain values of $x$ in rational expressions?
4. Can you explain how to find the domain of a rational expression in general?
5. How does factoring help simplify expressions before multiplication?

Tip: Always factor fully before canceling common terms in rational expressions to simplify accurately.