To simplify the rational expression \(\frac{2x + 4}{x^2 + 4x + 4} \times \frac{x^2 + 2x}{2x}\), let's break it down into steps.
Step 1: Factor the expressions
Numerator of the first term: \[ 2x + 4 = 2(x + 2) \]
Denominator of the first term: \[ x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2 \]
Numerator of the second term: \[ x^2 + 2x = x(x + 2) \]
Denominator of the second term: \[ 2x = 2x \]
Step 2: Rewrite the expression
Now substitute the factored forms back into the expression: \[ \frac{2(x + 2)}{(x + 2)^2} \times \frac{x(x + 2)}{2x} \]
Step 3: Cancel out common terms
Combining everything, we have: \[ \frac{2(x + 2) \cdot x(x + 2)}{(x + 2)^2 \cdot 2x} \]
We can cancel \(2x\) in the numerator and denominator, and cancel \(x + 2\) from the numerator and one \(x + 2\) in the denominator: \[ \frac{\cancel{2} \cdot \cancel{(x + 2)} \cdot x \cdot \cancel{(x + 2)}}{\cancel{(x + 2)} \cdot (x + 2) \cdot \cancel{2x}} = \frac{x}{x + 2} \]
Final Result
Thus, the fully simplified form of the given rational expression is: \[ \frac{x}{x + 2} \]
Assuming \(x \neq -2\) (to avoid division by zero) and \(x \neq 0\) (since there’s a \(2x\) in the denominator).
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