We are asked to simplify the given rational expression:

$\frac{6x^3 - 8x^2 y + 2xy^2}{18x^2 y - 2y^3}$

Step 1: Factor the numerator

The numerator is: $6x^3 - 8x^2 y + 2xy^2$ We can factor out the greatest common factor (GCF), which is $2x$: $2x(3x^2 - 4xy + y^2)$

Now, try to factor the quadratic expression $3x^2 - 4xy + y^2$. It can be factored as: $(3x - y)(x - y)$

Thus, the numerator becomes: $2x(3x - y)(x - y)$

Step 2: Factor the denominator

The denominator is: $18x^2 y - 2y^3$ We can factor out the GCF, which is $2y$: $2y(9x^2 - y^2)$ Now, recognize that $9x^2 - y^2$ is a difference of squares: $9x^2 - y^2 = (3x + y)(3x - y)$

Thus, the denominator becomes: $2y(3x + y)(3x - y)$

Step 3: Simplify the expression

Now, substitute the factored forms of the numerator and denominator into the expression: $\frac{2x(3x - y)(x - y)}{2y(3x + y)(3x - y)}$ We can cancel the common factors $2$ and $3x - y$ from both the numerator and the denominator: $\frac{x(x - y)}{y(3x + y)}$

Final Answer:

The simplified expression is: $\frac{x(x - y)}{y(3x + y)}$ Thus, the correct answer is:

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

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Would you like more details or have any questions about the steps? Here are 5 related questions to expand your understanding:

1. How do you recognize and factor the difference of squares?
2. Why is it important to factor out the greatest common factor first?
3. How would you simplify a similar rational expression with different terms?
4. What happens if you can't cancel any common factors in a rational expression?
5. How can you check that your factorization of a quadratic expression is correct?

Tip: Always factor out the greatest common factor (GCF) first—it simplifies the entire factoring process and helps avoid mistakes!