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The given expression is:

\[
\frac{1 - \frac{25}{x^2}}{1 + \frac{5}{x}}
\]

### Steps to simplify:

1. **Simplify the numerator**:
   The numerator is \( 1 - \frac{25}{x^2} \). To write this as a single fraction, express \( 1 \) as \( \frac{x^2}{x^2} \):

   \[
   1 - \frac{25}{x^2} = \frac{x^2 - 25}{x^2}
   \]

   Since \( x^2 - 25 \) is a difference of squares, it can be factored:

   \[
   \frac{x^2 - 25}{x^2} = \frac{(x - 5)(x + 5)}{x^2}
   \]

2. **Simplify the denominator**:
   The denominator is \( 1 + \frac{5}{x} \). Again, express \( 1 \) as \( \frac{x}{x} \):

   \[
   1 + \frac{5}{x} = \frac{x + 5}{x}
   \]

3. **Combine the expression**:
   Now we divide the simplified numerator by the simplified denominator:

   \[
   \frac{\frac{(x - 5)(x + 5)}{x^2}}{\frac{x + 5}{x}}
   \]

4. **Simplify the division**:
   When dividing by a fraction, multiply by its reciprocal:

   \[
   \frac{(x - 5)(x + 5)}{x^2} \times \frac{x}{x + 5}
   \]

   Cancel out the common factor \( x + 5 \):

   \[
   \frac{(x - 5)}{x}
   \]

Thus, the simplified expression is:

\[
\frac{x - 5}{x}
\]

Would you like more details on any step?

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Here are 5 related questions to consider:

1. How do you simplify expressions involving complex fractions?
2. What are the conditions for canceling terms in rational expressions?
3. How do you factor difference of squares?
4. What happens when the denominator equals zero in a rational expression?
5. How would the solution change if the signs in the expression were reversed?

**Tip**: When simplifying, always check if terms can be factored, as this often leads to cancellations.

Simplify \( \frac{1 - \frac{25}{x^2}}{1 + \frac{5}{x}} \)

This problem involves simplifying a rational expression using the difference of squares and the division of fractions. The given expression is \( \frac{1 - \frac{25}{x^2}}{1 + \frac{5}{x}} \), and the solution leads to the simplified form \( \frac{x - 5}{x} \). Suitable for students in Grades 9-11.

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