To simplify the rational expression
\[ \frac{2x + 4}{x^2 + 4x + 4} \times \frac{x^2 + 2x}{2x}, \]
we can start by factoring where possible.
- Factor the numerator and denominator:
- The numerator \(2x + 4\) can be factored as: \[ 2(x + 2). \]
- The quadratic \(x^2 + 4x + 4\) can be factored as: \[ (x + 2)(x + 2) = (x + 2)^2. \]
- The expression \(x^2 + 2x\) can also be factored as: \[ x(x + 2). \]
- The denominator \(2x\) is already simplified.
Putting these together, we have: \[ \frac{2(x + 2)}{(x + 2)^2} \times \frac{x(x + 2)}{2x}. \]
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Combine the fractions: Now, we can write this as: \[ \frac{2(x + 2) \cdot x(x + 2)}{(x + 2)^2 \cdot 2x}. \]
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Cancel common factors: In the numerator and denominator, we can cancel \(2\) from the numerator and denominator, and also \(x\) from the numerator and denominator:
- Cancel \(2\).
- Cancel one \(x + 2\) from the numerator and one \(x + 2\) from the denominator.
- Cancel \(x\).
We are left with: \[ \frac{x + 2}{x + 2}. \]
- Final simplification: Since \(x + 2\) cancels out, we have: \[
\]
Thus, the fully simplified expression is: \[ \boxed{1}, \quad \text{for } x \neq -2 \text{ (to avoid division by zero)}. \]
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