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To simplify the expression 

\[
\frac{x^2 - 25}{x^3 - 5x^2}
\]

let's proceed with factoring both the numerator and the denominator.

1. **Factor the Numerator**: 
   The numerator is \( x^2 - 25 \), which is a difference of squares. We can factor it as follows:
   \[
   x^2 - 25 = (x + 5)(x - 5)
   \]

2. **Factor the Denominator**: 
   The denominator is \( x^3 - 5x^2 \). We can factor out \( x^2 \) from both terms:
   \[
   x^3 - 5x^2 = x^2(x - 5)
   \]

Now, rewrite the expression with these factored forms:
\[
\frac{(x + 5)(x - 5)}{x^2(x - 5)}
\]

3. **Simplify by Canceling Common Terms**: 
   Notice that \( x - 5 \) is a common factor in both the numerator and the denominator. We can cancel \( x - 5 \) from both:
   \[
   \frac{(x + 5) \cancel{(x - 5)}}{x^2 \cancel{(x - 5)}} = \frac{x + 5}{x^2}
   \]

Thus, the simplified expression is:
\[
\frac{x + 5}{x^2}
\]

Would you like further details or have any questions?

Here are some related questions that expand on this concept:
1. How do you factor the difference of squares for different types of expressions?
2. What is the significance of canceling terms in a rational expression?
3. How can you identify when a polynomial can be factored out?
4. How would the solution change if there was a common factor in both terms of the denominator that wasn’t in the numerator?
5. What are some common factoring techniques for simplifying complex rational expressions?

**Tip:** Always check for common factors that can be canceled after factoring both the numerator and the denominator.

Learn how to simplify the rational expression (x^2 - 25) / (x^3 - 5x^2) by factoring and canceling common terms. This problem involves concepts such as the difference of squares and factoring out common factors, ideal for students in Grades 9-10.

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