To determine which expression is equivalent to [tex]\(\frac{2 x^2 + 2 x - 4}{2 x^2 - 4 x + 2}\)[/tex], we need to simplify the given fraction.
Starting with the fraction:
[tex]\[
\frac{2 x^2 + 2 x - 4}{2 x^2 - 4 x + 2}
\][/tex]
We factorize both the numerator and the denominator:
1. Factorize the Numerator:
The numerator is [tex]\(2x^2 + 2x - 4\)[/tex].
First, factor out the greatest common factor (GCF), which is 2:
[tex]\[
2(x^2 + x - 2)
\][/tex]
Next, factor the quadratic expression [tex]\(x^2 + x - 2\)[/tex]:
[tex]\[
x^2 + x - 2 = (x + 2)(x - 1)
\][/tex]
So, the numerator becomes:
[tex]\[
2(x + 2)(x - 1)
\][/tex]
2. Factorize the Denominator:
The denominator is [tex]\(2x^2 - 4x + 2\)[/tex].
First, factor out the GCF, which is 2:
[tex]\[
2(x^2 - 2x + 1)
\][/tex]
Next, factor the quadratic expression [tex]\(x^2 - 2x + 1\)[/tex]:
[tex]\[
x^2 - 2x + 1 = (x - 1)^2
\][/tex]
So, the denominator becomes:
[tex]\[
2(x - 1)^2
\][/tex]
3. Simplify the Fraction:
Putting the factored forms together, we have:
[tex]\[
\frac{2(x + 2)(x - 1)}{2(x - 1)^2}
\][/tex]
We can cancel the common factors in the numerator and the denominator:
[tex]\[
\frac{2 \cdot (x + 2) \cdot (x - 1)}{2 \cdot (x - 1) \cdot (x - 1)} = \frac{(x + 2)}{(x - 1)}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{x + 2}{x - 1}
\][/tex]
Therefore, the equivalent expression is [tex]\(\frac{x+2}{x-1}\)[/tex].
Among the given choices, the correct answer is:
[tex]\[
\boxed{\frac{x+2}{x-1}}
\][/tex]
