We are tasked with simplifying the given expression:

$\frac{x^2 - 3xy}{9y^2 - x^2}$

Let's first factor both the numerator and the denominator.

Step 1: Factor the numerator

The numerator is $x^2 - 3xy$. We can factor out $x$ from both terms:

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

Step 2: Factor the denominator

The denominator is $9y^2 - x^2$. This is a difference of squares, and it can be factored as:

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

Step 3: Rewrite the expression

Now substitute the factored forms back into the original expression:

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

Step 4: Simplify

Notice that $3y - x$ is the same as $-(x - 3y)$. So, we can rewrite the denominator as:

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

Cancel out the common term $x - 3y$ in both the numerator and denominator:

$\frac{x}{-(x + 3y)} = -\frac{x}{x + 3y}$

Final Answer:

The correct answer is $\boxed{B}$, which is $-\frac{x}{x + 3y}$.

Would you like any further details or clarifications on this solution? Here are some related questions:

1. How do you factor more complex polynomials?
2. What is the difference of squares and how is it applied in factorization?
3. Why is it valid to cancel out terms in the numerator and denominator?
4. How would this problem change if there were different exponents on $x$ or $y$?
5. Can this method be extended to rational functions with higher-degree polynomials?

Tip: Always check for factoring opportunities, such as common factors or patterns like difference of squares, to simplify expressions efficiently.