The derivative of the given function $f(x) = \frac{x^2 + 6x}{2x^2 + 4x + 4}$ is:

$f'(x) = \frac{(-4x - 4)(x^2 + 6x)}{(2x^2 + 4x + 4)^2} + \frac{2x + 6}{2x^2 + 4x + 4}$

Would you like a detailed step-by-step explanation of how this derivative was obtained?

Here are some related questions to explore this topic further:

1. How can you find the critical points of the function $f(x)$?
2. What is the behavior of $f(x)$ as $x \to \infty$?
3. How do you simplify complex rational expressions after taking derivatives?
4. Can you determine whether the function $f(x)$ has any inflection points?
5. How do you analyze the concavity of $f(x)$ using the second derivative?

Tip: When finding the derivative of a quotient, use the quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$.