Question

Simplify fully the rational expression 2x + 4/x² + 4x + 4 × x² + 2x/2x

Answer 1

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).