https://file.aiquickdraw.com/mathbot/file/1726266817196on35dgff.jpg

Let's break down the expression shown in the image:

\[
\frac{x^2 - 25}{x^2 + 5x} \div \frac{xy + 6x - 5y - 30}{5x - 15}
\]

To simplify this, we follow these steps:

### Step 1: Simplify the first fraction \(\frac{x^2 - 25}{x^2 + 5x}\).

1. **Factor the numerator**: 
   \[
   x^2 - 25 = (x - 5)(x + 5)
   \]

2. **Factor the denominator**: 
   \[
   x^2 + 5x = x(x + 5)
   \]

Thus, the first fraction simplifies to:
\[
\frac{(x - 5)(x + 5)}{x(x + 5)}
\]
Cancel out the \((x + 5)\) from the numerator and denominator:
\[
\frac{x - 5}{x}
\]

### Step 2: Simplify the second fraction \(\frac{xy + 6x - 5y - 30}{5x - 15}\).

1. **Factor the denominator**:
   \[
   5x - 15 = 5(x - 3)
   \]

2. **Group terms in the numerator for factoring**:
   \[
   xy + 6x - 5y - 30 = x(y + 6) - 5(y + 6)
   \]
   Factor the numerator:
   \[
   (x - 5)(y + 6)
   \]

Thus, the second fraction becomes:
\[
\frac{(x - 5)(y + 6)}{5(x - 3)}
\]

### Step 3: Apply the division rule (multiply by the reciprocal).

\[
\frac{x - 5}{x} \div \frac{(x - 5)(y + 6)}{5(x - 3)} = \frac{x - 5}{x} \times \frac{5(x - 3)}{(x - 5)(y + 6)}
\]

Cancel out the \((x - 5)\) from the numerator and denominator:
\[
\frac{5(x - 3)}{x(y + 6)}
\]

### Final Answer:
The simplified expression is:
\[
\frac{5(x - 3)}{x(y + 6)}
\]

Would you like more details or have any further questions?

Here are 5 related questions you can explore:
1. How do you multiply rational expressions?
2. How do you handle factoring polynomials?
3. What are the general steps for simplifying rational expressions?
4. How does the division of rational expressions differ from multiplication?
5. How do you cancel common factors in algebraic fractions?

**Tip**: Always factor polynomials completely before simplifying rational expressions!

Simplify the expression: (x^2 - 25) / (x^2 + 5x) ÷ (xy + 6x - 5y - 30) / (5x - 15)

Learn how to simplify the division of two rational expressions involving factoring and canceling common terms. The problem covers algebraic concepts such as factoring polynomials, handling rational expressions, and applying the division rule. Suitable for high school students.

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