Factor out the common factor from the numerator.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Factor out the common factor from the numerator.
$$6x^2 - 8xy + 2y^2 = 2(3x^2 - 4xy + y^2)$$
Step 2: Factor the quadratic expression in the parenthesis in the numerator.
$$3x^2 - 4xy + y^2 = (3x - y)(x - y)$$
So the numerator becomes:
$$2(3x - y)(x - y)$$
Step 3: Factor out the common factor from the denominator.
$$18x^2 - 2y^2 = 2(9x^2 - y^2)$$
Step 4: Factor the expression in the parenthesis in the denominator using the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$.
$$9x^2 - y^2 = (3x)^2 - y^2 = (3x - y)(3x + y)$$
So the denominator becomes:
$$2(3x - y)(3x + y)$$
Step 5: Substitute the factored expressions back into the fraction and simplify by canceling common factors.
$$\frac{2(3x - y)(x - y)}{2(3x - y)(3x + y)} = \frac{x - y}{3x + y}$$
The simplified expression is $\boxed{\frac{x - y}{3x + y}}$.
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