(6n + 6)/(n + 9) * (n²  + 6n - 27)/(6n + 6)

Let's simplify the given expression step by step.

The expression is:

\[
\frac{6n + 6}{n + 9} \times \frac{n^2 + 6n - 27}{6n + 6}
\]

### Step 1: Factorize the Numerators and Denominators

1. **Factorize \(6n + 6\):**

   \[
   6n + 6 = 6(n + 1)
   \]

2. **Factorize \(n^2 + 6n - 27\):**

   To factorize \(n^2 + 6n - 27\), we look for two numbers that multiply to \(-27\) (constant term) and add up to \(6\) (coefficient of \(n\)). Those numbers are \(9\) and \(-3\). Thus:

   \[
   n^2 + 6n - 27 = (n + 9)(n - 3)
   \]

3. **Rewrite the expression using these factorizations:**

   The expression becomes:

   \[
   \frac{6(n + 1)}{n + 9} \times \frac{(n + 9)(n - 3)}{6(n + 1)}
   \]

### Step 2: Cancel Common Factors

Notice that \(6(n + 1)\) appears in both the numerator and the denominator. Also, \((n + 9)\) appears in both the numerator and the denominator. We can cancel these common factors:

\[
\frac{\cancel{6(n + 1)}}{\cancel{n + 9}} \times \frac{\cancel{(n + 9)}(n - 3)}{\cancel{6(n + 1)}}
\]

What remains is:

\[
n - 3
\]

### Final Answer

The simplified form of the expression is:

\[
\boxed{n - 3}
\]

Would you like more details, or do you have any questions?

### Related Questions to Explore Further:

1. How do you factor quadratic expressions in general?
2. What are the rules for canceling terms in fractions?
3. Can every algebraic expression be simplified by factoring?
4. How do you determine the restrictions on the variables for rational expressions?
5. What are common mistakes to avoid when simplifying algebraic fractions?

**Tip:** Always check for common factors before multiplying or simplifying fractions. This can often simplify your work significantly!

(6n + 6)/(n + 9) * (n² + 6n - 27)/(6n + 6)

Learn how to simplify the expression (6n + 6)/(n + 9) * (n² + 6n - 27)/(6n + 6) step-by-step. This problem involves fraction simplification and algebraic factorization. Suitable for Grades 7-9.

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