Reduce the following rational expression to lowest terms: (6x^3 - 8x^2 y + 2 xy^2) / (18 x^2 y - 2 y^3)

a. [x(x + y)] / [y(3 x - y)]

b. [x(x + y)] / [y(3 x + y)]

c. [x(x - y)] / [y(3 x + y)]

d. [x(x - y)] / [y(3 x - y)]

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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!

Reduce the following rational expression to lowest terms: (6x^3 - 8x^2 y + 2 xy^2) / (18 x^2 y - 2 y^3)
a. [x(x + y)] / [y(3 x - y)]
b. [x(x + y)] / [y(3 x + y)]
c. [x(x - y)] / [y(3 x + y)]
d. [x(x - y)] / [y(3 x - y)]

Learn how to simplify the rational expression (6x^3 - 8x^2 y + 2xy^2) / (18x^2 y - 2y^3) using factoring techniques and the difference of squares. Follow a detailed, step-by-step solution suitable for high school students in Grades 9-12.

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