To solve this problem, we need to perform the division of two rational expressions and simplify the result.

Given:

$\frac{\frac{x^2 - 4x}{x^2 - 6x + 9}}{\frac{x - 4}{x - 3}}$

To divide two fractions, we multiply the first fraction by the reciprocal of the second:

$\frac{x^2 - 4x}{x^2 - 6x + 9} \times \frac{x - 3}{x - 4}$

Step-by-step Solution:

1. Factor each expression where possible:

   • $x^2 - 4x = x(x - 4)$
   • $x^2 - 6x + 9 = (x - 3)^2$
   • $x - 4$ and $x - 3$ are already factored.

   So, the expression becomes:

   $\frac{x(x - 4)}{(x - 3)^2} \times \frac{x - 3}{x - 4}$

2. Simplify by canceling common terms:

   • The $x - 4$ terms cancel each other.
   • One of the $x - 3$ in the denominator cancels with the $x - 3$ in the numerator.

   The simplified expression is:

   $\frac{x}{x - 3}$

Answer:

The correct answer is:

$\boxed{\frac{x}{x - 3}}$

This matches the third option from the list provided in the image.

Do you have any questions, or would you like a more detailed explanation of any step?

Related Questions:

1. How do you multiply and divide rational expressions?
2. What is the process to factor quadratic expressions?
3. How can you simplify complex rational expressions?
4. What are the common pitfalls when cancelling terms in rational expressions?
5. How do you identify restrictions on the variable in rational expressions?

Tip:

Always check for factors that can be canceled before multiplying or dividing rational expressions to simplify them more easily.