(6n + 6)/(n + 9) * (n ^ 2 + 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: Simplify the common factors

Notice that the numerator of the first fraction \((6n + 6)\) is also the denominator of the second fraction. These terms can cancel each other out:

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

### Step 2: Factor the remaining expression

Now, let's factor the quadratic expression \(n^2 + 6n - 27\) in the numerator:

To factor this, we look for two numbers that multiply to \(-27\) and add to \(6\). The factors are \(9\) and \(-3\):

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

So, the expression becomes:

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

### Step 3: Cancel the common factors

The \((n + 9)\) terms in the numerator and denominator can be canceled out:

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

### Final Answer:

The simplified expression is:

\[
n - 3
\]

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

Here are some related questions you might find helpful:

1. How do you factor quadratic expressions in general?
2. What are common mistakes to avoid when canceling terms in fractions?
3. How can you recognize and factor a difference of squares?
4. What other methods can be used to simplify rational expressions?
5. How would the process change if the original expression had additional terms?

**Tip:** Always check for common factors before multiplying or dividing expressions, as it can significantly simplify your work.

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) step-by-step with detailed explanations and examples. This problem covers algebraic concepts including rational expressions and factoring quadratic expressions, suitable for Grades 9-12.

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