Let's solve this step by step to find the quotient and determine which values of $x$ must be excluded from the domain.

Step 1: Simplify the given expression

We are given the expression:

$\frac{x^2 - 8x + 16}{x^2 + 4x - 21} \div \frac{-2x + 8}{x + 7}$

This is a division of two rational expressions. To simplify, we convert the division to multiplication by multiplying the first fraction by the reciprocal of the second fraction:

$\frac{x^2 - 8x + 16}{x^2 + 4x - 21} \times \frac{x + 7}{-2x + 8}$

Step 2: Factor each part

Now let's factor all the numerators and denominators:

1. Factor $x^2 - 8x + 16$: This is a perfect square trinomial, so it factors as: $(x - 4)^2$

2. Factor $x^2 + 4x - 21$: This is a quadratic expression, and it factors as: $(x + 7)(x - 3)$

3. Factor $-2x + 8$: Factor out the common factor of $-2$: $-2(x - 4)$

So the expression becomes:

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

Step 3: Simplify the expression

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

1. One factor of $(x - 4)$ cancels out from the numerator and denominator.
2. The $(x + 7)$ term cancels out from the numerator and denominator.

After canceling, we are left with:

$\frac{x - 4}{-2(x - 3)}$

Step 4: Final quotient in lowest terms

The simplified quotient is:

$\frac{x - 4}{-2(x - 3)}$

Step 5: Determine excluded values

Now, we need to find the values of $x$ that must be excluded from the domain. These are the values that make any denominator zero at any point during the simplification process.

• From $(x + 7)$, $x = -7$ must be excluded.
• From $(x - 3)$, $x = 3$ must be excluded.
• From $-2(x - 4)$, $x = 4$ must be excluded.

So, the values that must be excluded are $x = -7$, $x = 3$, and $x = 4$.

Final Answer:

The quotient in lowest terms is:

$\frac{x - 4}{-2(x - 3)}$

The values that must be excluded from the domain are:

• $x = -7$
• $x = 3$
• $x = 4$

Correct choices:

• A. $x = -7$
• C. $x = 3$
• D. $x = 4$

---

Would you like more details on this? Here are some related questions to explore further:

1. How do you factor a quadratic expression like $x^2 + 4x - 21$?
2. Why do we cancel terms in rational expressions?
3. What happens if you don't exclude values that make the denominator zero?
4. How do you handle division of fractions?
5. What are other techniques to simplify complex rational expressions?

Tip: Always check for common factors and excluded values before simplifying a rational expression.