-3x+21/-2x-4 × x^2-16/x-7 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)   \[x=-4\]
A
\[x=-4\]
(Choice B)   \[x=-2\]
B
\[x=-2\]
(Choice C)   \[x=4\]
C
\[x=4\]
(Choice D)   \[x=7\]
D
\[x=7\]

Let's begin by simplifying the given expression step by step:

The expression is:

\[
\frac{-3x + 21}{-2x - 4} \times \frac{x^2 - 16}{x - 7}
\]

### Step 1: Factor each term
1. **First fraction:**  
   \[
   \frac{-3x + 21}{-2x - 4}
   \]  
   Factor out the greatest common factor (GCF) from the numerator and denominator:
   \[
   \frac{-3(x - 7)}{-2(x + 2)}
   \]
   
2. **Second fraction:**  
   \[
   \frac{x^2 - 16}{x - 7}
   \]  
   Notice that \(x^2 - 16\) is a difference of squares, which can be factored as:
   \[
   \frac{(x - 4)(x + 4)}{x - 7}
   \]

### Step 2: Multiply the two fractions
Now multiply the factored fractions:
\[
\frac{-3(x - 7)}{-2(x + 2)} \times \frac{(x - 4)(x + 4)}{x - 7}
\]

Since \((x - 7)\) appears in both the numerator and denominator, we can cancel it out:
\[
\frac{-3}{-2(x + 2)} \times (x - 4)(x + 4)
\]

This simplifies to:
\[
\frac{3(x - 4)(x + 4)}{2(x + 2)}
\]

### Step 3: Exclude values of \(x\) that would make any denominator zero
Now, we need to determine which values of \(x\) make any denominator zero:

1. From the first fraction's denominator, \(-2(x + 2) = 0\) when \(x = -2\).
2. From the second fraction's denominator, \(x - 7 = 0\) when \(x = 7\).

Thus, we must exclude \(x = -2\) and \(x = 7\) from the domain.

### Final Answer:
The product in its lowest terms is:
\[
\frac{3(x - 4)(x + 4)}{2(x + 2)}
\]

The values of \(x\) that must be excluded are \(x = -2\) and \(x = 7\).

**Correct choices:**
- (Choice B) \(x = -2\)
- (Choice D) \(x = 7\)

Would you like a detailed explanation of any step? Here are 5 related questions to consider:

1. How do you factor quadratic expressions like \(x^2 - 16\)?
2. Why do we cancel terms from the numerator and denominator when simplifying fractions?
3. What happens if you don't exclude values that make the denominator zero?
4. How do we handle the multiplication of rational expressions?
5. What is the significance of factoring in simplifying algebraic expressions?

**Tip:** Always check for common factors before multiplying or dividing expressions to simplify your work.

-3x+21/-2x-4 × x^2-16/x-7 What is the product in lowest terms? What values of [x] must we exclude from the domains of the expressions?

Learn how to simplify the rational expression (-3x + 21)/(-2x - 4) × (x^2 - 16)/(x - 7) and find the product in its lowest terms. This problem also explores factoring, canceling common terms, and determining which values must be excluded from the domain. Suitable for students in Grades 9-11.

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