X^2-8x+16/x^2+4x-21/-2x+8/x+7 What is the quotient 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)   \[x=-7\]
A
\[x=-7\]
(Choice B)   \[x=0\]
B
\[x=0\]
(Choice C)   \[x=3\]
C
\[x=3\]
(Choice D)   \[x=4\]
D
\[x=4\]

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.

X^2-8x+16/x^2+4x-21/-2x+8/x+7 What is the quotient in lowest terms? What values of x must we exclude from the domains of the expressions? Choose all answers that apply: A) x=-7, B) x=0, C) x=3, D) x=4

Learn how to simplify the rational expression X^2-8x+16 / (x^2 + 4x - 21) ÷ (-2x + 8) / (x + 7) and find the quotient in lowest terms. This problem covers key algebraic concepts such as factoring quadratics and determining excluded values. Suitable for students in Grades 9-11.

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