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: $x = 6, x = 7$

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!