Let's begin by analyzing the rational expression:

$\frac{x^2 + 2x}{x^2 + 4x + 4}$

Step 1: Factor the numerator and denominator

Numerator:

The numerator is $x^2 + 2x$. We can factor out an $x$:

$x^2 + 2x = x(x + 2)$

Denominator:

The denominator is $x^2 + 4x + 4$. This is a perfect square trinomial and can be factored as:

$x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2$

Step 2: Write the expression in factored form

Now that both the numerator and denominator are factored, we have:

$\frac{x(x + 2)}{(x + 2)^2}$

Step 3: Simplify

We can cancel one factor of $(x + 2)$ from both the numerator and denominator, as long as $x \neq -2$ (since $x = -2$ would make the denominator zero, and division by zero is undefined).

$\frac{x}{x + 2} \quad \text{(for } x \neq -2\text{)}$

Thus, the simplified form of the rational expression is:

$\frac{x}{x + 2}$

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Would you like further details on any of these steps or have any questions?

Here are 5 related questions to explore:

1. What is the domain of the simplified rational expression?
2. How would the rational expression change if the numerator was $x^2 + 3x$?
3. Can the expression $\frac{x}{x+2}$ be simplified further for specific values of $x$?
4. How can rational expressions be added or subtracted?
5. What happens if $x = -2$ in this context?

Tip: Always check for values that make the denominator zero to avoid undefined expressions.