(6n + 6)/(n + 9) * (n ^ 2 + 6n - 27)/(6n + 6)

The given expression is:

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

### Step 1: Simplify the expression
Notice that the term \(\frac{6n + 6}{6n + 6}\) appears in the expression, which simplifies to 1:

\[
\frac{6n + 6}{n + 9} \times \frac{n^2 + 6n - 27}{6n + 6} = \frac{1}{n + 9} \times (n^2 + 6n - 27)
\]

### Step 2: Simplify further
Now, the expression reduces to:

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

### Step 3: Factorize the quadratic expression \(n^2 + 6n - 27\)
To factor \(n^2 + 6n - 27\), we look for two numbers that multiply to \(-27\) and add to \(6\). These numbers are \(9\) and \(-3\), so:

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

### Step 4: Substitute the factorized form back into the expression
Substitute this into the expression:

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

### Step 5: Cancel out the common terms
The \(n + 9\) terms cancel out:

\[
n - 3
\]

Thus, the simplified expression is:

\[
\boxed{n - 3}
\]

Would you like more details or have any questions about this process?

### Related Questions
1. How do you factor quadratic equations in general?
2. What is the process for simplifying complex fractions?
3. Can you explain the difference between factoring and expanding expressions?
4. How do you solve for \(n\) in equations like this one?
5. What are some common mistakes to avoid when simplifying algebraic expressions?

**Tip:** Always check for common factors that can simplify the expression early in the process to make the calculation easier.

The query submitted by the user: (6n + 6)/(n + 9) * (n^2 + 6n - 27)/(6n + 6)

Learn how to simplify the rational expression (6n + 6)/(n + 9) * (n^2 + 6n - 27)/(6n + 6) by factoring and canceling common terms. This solution is ideal for students in Grades 9-11 and covers essential algebraic concepts like factorization and simplifying rational expressions.

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