https://file.aiquickdraw.com/mathbot/file/1729094920614h1qdszby.jpg

The expression given in the image is:

\[
\frac{7x^2 - 21x}{x^2 + 2x - 15}
\]

### Step 1: Factor the numerator and denominator

**Numerator:**  
\( 7x^2 - 21x \)  
Factor out the common factor, which is \( 7x \):
\[
7x(x - 3)
\]

**Denominator:**  
\( x^2 + 2x - 15 \)  
We need to factor this quadratic expression. Look for two numbers that multiply to give \( -15 \) and add to give \( 2 \). The numbers are \( 5 \) and \( -3 \), so we can factor the quadratic as:
\[
(x + 5)(x - 3)
\]

### Step 2: Simplify the expression
Now we have:
\[
\frac{7x(x - 3)}{(x + 5)(x - 3)}
\]

Cancel out the common factor \( (x - 3) \) from both the numerator and denominator, leaving:
\[
\frac{7x}{x + 5}
\]

This is the simplified form of the expression.

Would you like further clarification on any step?

### Related questions:
1. How do you factor quadratics efficiently?
2. What are the conditions under which you can cancel terms in a rational expression?
3. Can you explain how to handle cases where a factor equals zero in rational expressions?
4. How would the simplification change if the expression involved higher powers of \( x \)?
5. What if the denominator had no real factors—how would the simplification be approached?

**Tip:** Always check for restrictions after simplifying. In this case, \( x \neq 3 \) and \( x \neq -5 \) to avoid division by zero.

Simplify fully: (7x^2 - 21x) / (x^2 + 2x - 15)

Learn how to simplify the rational expression (7x^2 - 21x) / (x^2 + 2x - 15) by factoring both the numerator and denominator. This problem demonstrates key algebraic concepts, including factoring quadratic expressions and simplifying rational expressions. Suitable for students in Grades 9-10.

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