Let's work through the two questions step by step.

Given Expression:

$\dfrac{-3x+21}{-2x-4} \cdot \dfrac{x^2-16}{x-7}$

Step 1: Simplify the Expression

Factor the terms where possible:

• Numerator of the first fraction: $-3x + 21 = -3(x - 7)$
• Denominator of the first fraction: $-2x - 4 = -2(x + 2)$
• Numerator of the second fraction: $x^2 - 16 = (x - 4)(x + 4) \quad \text{(difference of squares)}$
• Denominator of the second fraction: $x - 7 \quad \text{(already factored)}$

So the expression becomes: $\dfrac{-3(x - 7)}{-2(x + 2)} \cdot \dfrac{(x - 4)(x + 4)}{x - 7}$

Step 2: Cancel Common Factors

Now, we can cancel out common factors between the numerator and denominator.

• Factor $x - 7$ appears in both the numerator and denominator, so we can cancel it out.

The expression now simplifies to: $\dfrac{-3(x + 4)}{-2(x + 2)}$

Step 3: Simplify Further

The remaining expression is: $\dfrac{-3(x + 4)}{-2(x + 2)} = \dfrac{3(x + 4)}{2(x + 2)}$

Answer to Question 1: What is the product in lowest terms?

The product of the two rational expressions in lowest terms is: $\dfrac{3(x + 4)}{2(x + 2)}$

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Step 4: Identify the Excluded Values (Domain Restrictions)

To find the excluded values for $x$, we need to examine the original expression's denominators. The expression is undefined when any denominator is zero, so we solve for $x$ in the following cases:

1. First denominator: $-2x - 4 = 0 \quad \Rightarrow \quad -2x = 4 \quad \Rightarrow \quad x = -2$
2. Second denominator: $x - 7 = 0 \quad \Rightarrow \quad x = 7$

Thus, the values of $x$ that make the denominator zero (and must be excluded from the domain) are $x = -2$ and $x = 7$.

Answer to Question 2: What values of $x$ must we exclude from the domains of the expressions?

The values to exclude are:

• $x = -2$
• $x = 7$

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Summary of Answers:

1. The product in lowest terms is: $\dfrac{3(x + 4)}{2(x + 2)}$
2. The values of $x$ to exclude from the domain are:
   • $x = -2$
   • $x = 7$

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Would you like further details on any of the steps or concepts? Here are some related questions:

1. How do we handle the cancellation of terms in rational expressions?
2. Can we ever cancel terms involving addition or subtraction in rational expressions?
3. How can we check if an expression is fully simplified?
4. What is the difference between a factor and a common factor in algebraic expressions?
5. What are the key concepts to keep in mind when solving rational expressions?

Tip: Always check for common factors before canceling terms to ensure you don't overlook any restrictions.