Let us solve the problem step-by-step to simplify the given expression.

Expression:

$\frac{\frac{1}{x^2 - 9} - \frac{2}{x}}{\frac{x - 3}{x^2 - x}}$

Step 1: Simplify the denominator $x^2 - x$

Factor $x^2 - x$: $x^2 - x = x(x - 1)$

Step 2: Simplify the numerator $\frac{1}{x^2 - 9} - \frac{2}{x}$

Factor $x^2 - 9$ as $(x - 3)(x + 3)$. Rewrite: $\frac{1}{x^2 - 9} - \frac{2}{x} = \frac{1}{(x - 3)(x + 3)} - \frac{2}{x}$

The least common denominator (LCD) is: $x(x - 3)(x + 3)$

Rewrite the fractions with the LCD: $\frac{1}{(x - 3)(x + 3)} = \frac{x}{x(x - 3)(x + 3)}, \quad \frac{2}{x} = \frac{2(x - 3)(x + 3)}{x(x - 3)(x + 3)}$

Combine the fractions: $\frac{1}{(x - 3)(x + 3)} - \frac{2}{x} = \frac{x - 2(x - 3)(x + 3)}{x(x - 3)(x + 3)}$

Expand and simplify the numerator: $x - 2(x - 3)(x + 3) = x - 2(x^2 - 9) = x - 2x^2 + 18$

Thus: $\frac{1}{(x - 3)(x + 3)} - \frac{2}{x} = \frac{-2x^2 + x + 18}{x(x - 3)(x + 3)}$

Step 3: Combine numerator and denominator

Now consider the full expression: $\frac{\frac{-2x^2 + x + 18}{x(x - 3)(x + 3)}}{\frac{x - 3}{x(x - 1)}}$

To divide fractions, multiply by the reciprocal: $\frac{\frac{-2x^2 + x + 18}{x(x - 3)(x + 3)}}{\frac{x - 3}{x(x - 1)}} = \frac{-2x^2 + x + 18}{x(x - 3)(x + 3)} \cdot \frac{x(x - 1)}{x - 3}$

Cancel common factors:

1. $x$ cancels.
2. $x - 3$ cancels.

Result: $\frac{-2x^2 + x + 18}{(x + 3)} \cdot (x - 1)$

Multiply the numerator: $(-2x^2 + x + 18)(x - 1)$

Expand: $(-2x^2 + x + 18)(x - 1) = -2x^3 + 2x^2 + x^2 - x + 18x - 18 = -2x^3 + 3x^2 + 17x - 18$

Thus the simplified expression is: $\frac{-2x^3 + 3x^2 + 17x - 18}{x + 3}$

Would you like a detailed explanation of any specific step?

Questions to Explore:

1. How do you factorize quadratic and cubic polynomials?
2. Why is finding the LCD necessary when combining fractions?
3. How do you multiply and simplify rational expressions?
4. What methods help in verifying the correctness of the simplification?
5. What are the common mistakes when working with complex rational expressions?

Tip: Always factorize completely before simplifying rational expressions; this helps identify common terms that can cancel.