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Let's break down the problem step by step.

### Given Expression:
You need to simplify the rational division:

\[
\frac{-4x}{x - 7} \div \frac{x^2 - 12x + 36}{2x - 14}
\]

### Step 1: Change Division to Multiplication

Dividing by a fraction is equivalent to multiplying by its reciprocal. So, we rewrite the expression as:

\[
\frac{-4x}{x - 7} \times \frac{2x - 14}{x^2 - 12x + 36}
\]

### Step 2: Factorize Terms

- Factorize \( x^2 - 12x + 36 \):
  \[
  x^2 - 12x + 36 = (x - 6)(x - 6)
  \]

- Factorize \( 2x - 14 \):
  \[
  2x - 14 = 2(x - 7)
  \]

Now, the expression becomes:

\[
\frac{-4x}{x - 7} \times \frac{2(x - 7)}{(x - 6)(x - 6)}
\]

### Step 3: Simplify

- Cancel the common factor \( x - 7 \) from both the numerator and the denominator:
  \[
  \frac{-4x}{1} \times \frac{2}{(x - 6)(x - 6)}
  \]
  This simplifies to:
  \[
  \frac{-8x}{(x - 6)^2}
  \]

### Step 4: Domain Restrictions

- The domain excludes values where the denominator is zero. From the expression \( (x - 6)^2 \), we know that \( x = 6 \) must be excluded from the domain.
- Additionally, from the original denominators \( x - 7 \) and \( 2x - 14 = 2(x - 7) \), we exclude \( x = 7 \).

Thus, the values that must be excluded from the domain are \( x = 6 \) and \( x = 7 \).

### Final Answers:
1. The quotient in lowest terms is:
   \[
   \frac{-8x}{(x - 6)^2}
   \]

2. The values of \( x \) to exclude from the domain are:
   \[
   x = 6, x = 7
   \]

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Would you like a more detailed explanation of any step? Here are some related questions you can explore:

1. What happens when dividing two rational expressions?
2. How do you find the domain of a rational expression?
3. How do you factor quadratic expressions like \( x^2 - 12x + 36 \)?
4. Why does division by zero affect the domain?
5. What is the significance of canceling terms in rational expressions?

**Tip:** Always check for factors in both the numerator and denominator that can simplify your rational expressions!

Simplify the rational division: (-4x / (x - 7)) ÷ ((x^2 - 12x + 36) / (2x - 14)). What is the quotient in lowest terms? What values of x must be excluded from the domain?

Learn how to simplify the rational division of (-4x / (x - 7)) ÷ ((x^2 - 12x + 36) / (2x - 14)) with a step-by-step solution. This problem covers key algebra concepts such as factoring, domain restrictions, and simplifying complex rational expressions.

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