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.