We need to simplify two expressions. 1. $\frac{x^2 - 10x + 25}{x^2 - 25}$

1. Problem Description

We need to simplify two expressions.

1. $\frac{x^2 - 10x + 25}{x^2 - 25}$

2. $\frac{x^2 - 3x}{x^2 - x - 2} \times \frac{x-2}{x^2 - 6x + 9}$

2. Solution Steps

Problem 1: $\frac{x^2 - 10x + 25}{x^2 - 25}$

* Factor the numerator and denominator.

$x^2 - 10x + 25 = (x-5)(x-5) = (x-5)^2$

$x^2 - 25 = (x-5)(x+5)$

* Rewrite the expression.

$\frac{(x-5)^2}{(x-5)(x+5)} = \frac{(x-5)(x-5)}{(x-5)(x+5)}$

* Cancel the common factor $(x-5)$.

$\frac{x-5}{x+5}$

Problem 2: $\frac{x^2 - 3x}{x^2 - x - 2} \times \frac{x-2}{x^2 - 6x + 9}$

* Factor each part of the expression.

$x^2 - 3x = x(x-3)$

$x^2 - x - 2 = (x-2)(x+1)$

$x^2 - 6x + 9 = (x-3)(x-3) = (x-3)^2$

* Rewrite the expression with factored forms.

$\frac{x(x-3)}{(x-2)(x+1)} \times \frac{x-2}{(x-3)^2} = \frac{x(x-3)(x-2)}{(x-2)(x+1)(x-3)(x-3)}$

* Cancel the common factors $(x-3)$ and $(x-2)$.

$\frac{x}{(x+1)(x-3)}$

* Simplify.

$\frac{x}{x^2 - 3x + x - 3} = \frac{x}{x^2 - 2x - 3}$

3. Final Answer

Problem 1: $\frac{x-5}{x+5}$

Problem 2: $\frac{x}{(x+1)(x-3)} = \frac{x}{x^2 - 2x - 3}$