Question

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

Answer 1

To simplify the rational expression

\[
\frac{2x + 4}{x^2 + 4x + 4} \times \frac{x^2 + 2x}{2x},
\]

we can start by factoring where possible.

1. **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:
\[
\frac{2(x + 2)}{(x + 2)^2} \times \frac{x(x + 2)}{2x}.
\]

2. **Combine the fractions:**
Now, we can write this as:
\[
\frac{2(x + 2) \cdot x(x + 2)}{(x + 2)^2 \cdot 2x}.
\]

3. **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:
\[
\frac{x + 2}{x + 2}.
\]

4. **Final simplification:**
Since \(x + 2\) cancels out, we have:
\[
1.
\]

Thus, the fully simplified expression is:
\[
\boxed{1}, \quad \text{for } x \neq -2 \text{ (to avoid division by zero)}.
\]