-4x-32/x^2-12x+36 × x^2-9x+18/3x+24 What is the product in lowest terms? What values of \[x\] must we exclude from the domains of the expressions?
Choose all answers that apply:
Choose all answers that apply:
(Choice A)   \[x=-8\]
A
\[x=-8\]
(Choice B)   \[x=0\]
B
\[x=0\]
(Choice C)   \[x=3\]
C
\[x=3\]
(Choice D)   \[x=6\]
D
\[x=6\]

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.

-4x - 32 / (x^2 - 12x + 36) × (x^2 - 9x + 18) / (3x + 24) What is the product in lowest terms? What values of x must we exclude from the domains of the expressions? Choose all answers that apply: (A) x=-8, (B) x=0, (C) x=3, (D) x=6

Learn how to simplify the rational expression (-4x - 32)/(x^2 - 12x + 36) × (x^2 - 9x + 18)/(3x + 24) by factoring and canceling terms. Understand how to identify excluded values like x = -8 and x = 6 that make the denominator zero. Suitable for high school students studying Algebra.

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