Simplify the expression (4x² - 16y²) / (6x² - 8xy - 8y²)

The numerator is $4x^2 - 16y^2$. We can factor out the common factor of 4, and then recognize it as a difference of squares ($a^2 - b^2 = (a-b)(a+b)$).

$$4x^2 - 16y^2 = 4(x^2 - 4y^2)$$ $$4(x^2 - (2y)^2) = 4(x-2y)(x+2y)$$

The denominator is $6x^2 - 8xy - 8y^2$. First, factor out the common factor of 2.

$$6x^2 - 8xy - 8y^2 = 2(3x^2 - 4xy - 4y^2)$$

Now, factor the quadratic expression $3x^2 - 4xy - 4y^2$. We look for two numbers that multiply to $3 \times (-4) = -12$ and add to $-4$. These numbers are $-6$ and 2.

$$3x^2 - 6xy + 2xy - 4y^2$$

Group the terms and factor:

$$(3x^2 - 6xy) + (2xy - 4y^2)$$ $$3x(x - 2y) + 2y(x - 2y)$$ $$(3x + 2y)(x - 2y)$$

So, the fully factored denominator is:

$$2(3x + 2y)(x - 2y)$$

Substitute the factored forms of the numerator and denominator back into the original expression:

$$\frac{4(x-2y)(x+2y)}{2(3x+2y)(x-2y)}$$

Cancel out the common factors $(x-2y)$ from the numerator and denominator, and simplify the numerical coefficients $\frac{4}{2} = 2$.

$$\frac{2(x+2y)}{3x+2y}$$

The final answer is $\boxed{\frac{2(x+2y)}{3x+2y}}$.

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