Question

which expression is equivalent to \\(\frac{2x^2 + 2x - 4}{2x^2 - 4x + 2}\\)?
\\(\frac{x + 2}{x - 1}\\)
\\(x + 2\\)
\\(-2\\)
\\(\frac{x + 2}{x - 2}\\)

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(2x^2 + 2x - 4\). We can factor out a 2 first: \(2(x^2 + x - 2)\). Then factor the quadratic: \(x^2 + x - 2=(x + 2)(x - 1)\), so the numerator becomes \(2(x + 2)(x - 1)\).

Next, factor the denominator \(2x^2 - 4x + 2\). Factor out a 2: \(2(x^2 - 2x + 1)\). The quadratic \(x^2 - 2x + 1=(x - 1)^2\), so the denominator becomes \(2(x - 1)^2\).

Step2: Simplify the fraction

Now we have the fraction \(\frac{2(x + 2)(x - 1)}{2(x - 1)^2}\). We can cancel out the common factors. The 2 in the numerator and denominator cancels, and one \((x - 1)\) in the numerator and denominator cancels, leaving \(\frac{x + 2}{x - 1}\).

Answer:

A. \(\frac{x + 2}{x - 1}\)