https://file.aiquickdraw.com/mathbot/file/172588989722760rc1z8h.jpg

The expression in the image is:

\[
\frac{r^2 + 4r - 21}{r^2 + 6r - 7} \cdot \frac{r + 1}{3r - 9}
\]

Let's simplify this step-by-step.

### Step 1: Factor the numerators and denominators
We need to factor both the quadratic expressions and simplify the terms.

1. **Factor** \( r^2 + 4r - 21 \):
   \[
   r^2 + 4r - 21 = (r + 7)(r - 3)
   \]

2. **Factor** \( r^2 + 6r - 7 \):
   \[
   r^2 + 6r - 7 = (r + 7)(r - 1)
   \]

3. The second fraction involves linear expressions:
   \[
   \frac{r + 1}{3r - 9}
   \]
   The denominator \( 3r - 9 \) can be factored as:
   \[
   3r - 9 = 3(r - 3)
   \]

### Step 2: Substitute the factored forms
We now substitute the factored forms into the original expression:

\[
\frac{(r + 7)(r - 3)}{(r + 7)(r - 1)} \cdot \frac{r + 1}{3(r - 3)}
\]

### Step 3: Simplify the expression
Cancel out common factors:
- \( (r + 7) \) appears in both the numerator and denominator, so we cancel it.
- \( (r - 3) \) also appears in both the numerator and denominator, so we cancel it.

This leaves:

\[
\frac{1}{r - 1} \cdot \frac{r + 1}{3}
\]

Now, multiply the remaining terms:

\[
\frac{(r + 1)}{3(r - 1)}
\]

### Final Answer:
The simplified form of the given expression is:

\[
\frac{r + 1}{3(r - 1)}
\]

---

Would you like a more detailed explanation or have any questions on specific steps?

### 5 Related Questions:
1. How do you factor quadratic expressions?
2. What are the general steps for simplifying algebraic fractions?
3. How do you handle cancellation of terms in rational expressions?
4. What other types of algebraic expressions can be simplified in a similar way?
5. How can you check if your factored form is correct?

**Tip**: When simplifying expressions with fractions, always check for common factors to cancel and avoid leaving terms that could further simplify the expression!

Simplify the algebraic expression: (r^2 + 4r - 21) / (r^2 + 6r - 7) * (r + 1) / (3r - 9).

Learn how to simplify the rational expression (r^2 + 4r - 21) / (r^2 + 6r - 7) * (r + 1) / (3r - 9) by factoring and canceling common factors. Suitable for students in Grades 8-10, this problem covers key algebraic concepts like factoring quadratics and simplifying rational expressions.

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