-4x-12/x-7÷x^2-12x+36/2x-14 What is the quotient in lowest terms?

We are asked to find the quotient of the expression:

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

### Step 1: Rewrite the division as multiplication by the reciprocal
We know that dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite the expression as:

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

### Step 2: Factor the expressions
Let's factor each part of the expression where possible:

1. **Factor \(-4x - 12\):**
   \[
   -4x - 12 = -4(x + 3)
   \]

2. **Factor \(x - 7\):**
   This is already in its simplest form.

3. **Factor \(2x - 14\):**
   \[
   2x - 14 = 2(x - 7)
   \]

4. **Factor \(x^2 - 12x + 36\) (this is a perfect square trinomial):**
   \[
   x^2 - 12x + 36 = (x - 6)^2
   \]

### Step 3: Substitute the factored forms back into the expression
Now that we have factored each part, substitute them back into the original equation:

\[
\frac{-4(x + 3)}{x - 7} \times \frac{2(x - 7)}{(x - 6)^2}
\]

### Step 4: Simplify by canceling common factors
Notice that \(x - 7\) appears in both the numerator and denominator, so we can cancel it out:

\[
\frac{-4(x + 3)}{1} \times \frac{2}{(x - 6)^2}
\]

Now, multiply the numerators and denominators:

\[
\frac{-4(x + 3) \times 2}{(x - 6)^2}
\]

This simplifies to:

\[
\frac{-8(x + 3)}{(x - 6)^2}
\]

### Step 5: Final simplified quotient
The quotient in its lowest terms is:

\[
\boxed{\frac{-8(x + 3)}{(x - 6)^2}}
\]

This is the final simplified expression.

Do you want any further clarification or details?

Here are 5 questions that might expand your understanding:
1. How do you factor a trinomial like \(x^2 - 12x + 36\)?
2. What are the steps to simplify rational expressions?
3. Why does dividing by a fraction turn into multiplying by the reciprocal?
4. How do you know when you can cancel terms in a fraction?
5. What are common types of factoring methods used in algebra?

**Tip:** When simplifying rational expressions, always look for common factors in the numerator and denominator to cancel out before multiplying.

4x-12/x-7 ÷ x^2-12x+36/2x-14 What is the quotient in lowest terms?

Learn how to simplify the rational expression 4x-12/x-7 ÷ x^2-12x+36/2x-14. This problem involves key algebraic concepts such as factoring, multiplying by the reciprocal, and simplifying rational expressions. Suitable for students in Grades 9-11.

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