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Let's simplify the given expression step-by-step:

The expression is:
\[
\left( x + \frac{9}{x - 6} \right) \div \left( \frac{x^2 - 9}{x^2 - 3x - 18} \right)
\]

### Step 1: Rewrite the Division as Multiplication
Dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, we can rewrite the expression as:
\[
\left( x + \frac{9}{x - 6} \right) \times \frac{x^2 - 3x - 18}{x^2 - 9}
\]

### Step 2: Simplify Each Part
#### Part 1: Simplify \( x + \frac{9}{x - 6} \)
Rewrite \( x + \frac{9}{x - 6} \) with a common denominator:
\[
x + \frac{9}{x - 6} = \frac{x(x - 6) + 9}{x - 6} = \frac{x^2 - 6x + 9}{x - 6} = \frac{(x - 3)^2}{x - 6}
\]

#### Part 2: Simplify \( \frac{x^2 - 3x - 18}{x^2 - 9} \)
Factor the numerator and denominator:
\[
x^2 - 3x - 18 = (x - 6)(x + 3)
\]
\[
x^2 - 9 = (x - 3)(x + 3)
\]
Thus,
\[
\frac{x^2 - 3x - 18}{x^2 - 9} = \frac{(x - 6)(x + 3)}{(x - 3)(x + 3)} = \frac{x - 6}{x - 3}
\]

### Step 3: Combine the Expressions
Now, we substitute back:
\[
\frac{(x - 3)^2}{x - 6} \times \frac{x - 6}{x - 3}
\]
The \( x - 6 \) terms cancel, and we are left with:
\[
\frac{(x - 3)^2}{x - 3} = x - 3
\]

### Final Answer
The simplified form of the expression is:
\[
\boxed{x - 3}
\]

### Answer Choice
The correct answer is:
**A) \( x - 3 \)**

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How can division of fractions be simplified by using multiplication?
2. What are the steps to factor quadratic expressions like \( x^2 - 3x - 18 \)?
3. Why is it important to cancel out terms in fractions carefully?
4. What are some common mistakes to avoid when simplifying complex expressions?
5. How does finding a common denominator help in combining fractions?

**Tip:** Always double-check that you have fully simplified all terms before selecting an answer.

(x + 9/(x - 6)) : ((x^2 - 9)/(x^2 - 3x - 18)) ifadesinin sadeleştirilmiş biçimi aşağıdakilerden hangisidir?

Learn how to simplify the expression (x + 9/(x - 6)) : ((x^2 - 9)/(x^2 - 3x - 18)) with detailed steps. This algebra problem covers key concepts such as rational expressions, factoring quadratics, and multiplying by the reciprocal. Suitable for high school students.

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