We are asked to divide two rational expressions:

$\frac{x^2 - 25}{6x} \div \frac{5 - x}{3xy}$

Step 1: Rewrite division as multiplication

When dividing by a fraction, we multiply by its reciprocal:

$\frac{x^2 - 25}{6x} \times \frac{3xy}{5 - x}$

Step 2: Factor where possible

We can factor the numerator $x^2 - 25$, which is a difference of squares:

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

So the expression becomes:

$\frac{(x - 5)(x + 5)}{6x} \times \frac{3xy}{5 - x}$

Step 3: Simplify the expression

Notice that $5 - x = -(x - 5)$. So we can replace $5 - x$ with $-(x - 5)$:

$\frac{(x - 5)(x + 5)}{6x} \times \frac{3xy}{-(x - 5)}$

Now, we can cancel out the common factor of $(x - 5)$:

$\frac{(x + 5)}{6x} \times \frac{3xy}{-1}$

Step 4: Multiply the fractions

Now, multiply the numerators and denominators:

$\frac{(x + 5)(3xy)}{6x(-1)}$

This simplifies to:

$\frac{-3xy(x + 5)}{6x}$

Step 5: Cancel common factors

We can cancel $x$ from the numerator and denominator:

$\frac{-3y(x + 5)}{6}$

Finally, simplify the constants:

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

Final Answer:

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

Let me know if you'd like any clarifications!

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Here are 5 related questions:

1. How would the solution change if the expression involved a difference instead of a product?
2. Can you explain how to factor higher-degree polynomials similarly?
3. What would happen if there were additional variables in the expression?
4. How do you handle division of rational expressions with exponents greater than 2?
5. What other types of factoring could be used to simplify more complex terms?

Tip: Always look for common factors in both the numerator and denominator when simplifying rational expressions!