Answer :
To simplify the expression \(\frac{2x^2 - 2}{-x + 1}\), follow these steps:
1. Factor the Numerator: Start by factoring the numerator \(2x^2 - 2\). Notice that both terms in the numerator share a common factor of 2:
[tex]\[ 2x^2 - 2 = 2(x^2 - 1) \][/tex]
2. Recognize a Difference of Squares: Recognize that \(x^2 - 1\) is a difference of squares, which can be factored further:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
Substituting this back into the expression, we have:
[tex]\[ 2(x^2 - 1) = 2(x - 1)(x + 1) \][/tex]
3. Rewrite the Expression: Substitute the factored form into the original expression:
[tex]\[ \frac{2(x - 1)(x + 1)}{-x + 1} \][/tex]
4. Simplify by Canceling Common Factors: Observe that \((x - 1)\) and \((-x + 1)\) are related. Note that \((-x + 1)\) can be rewritten as \(-(x - 1)\). So, the expression becomes:
[tex]\[ \frac{2(x - 1)(x + 1)}{-(x - 1)} = -2(x + 1) \][/tex]
Here, the \((x - 1)\) terms cancel out, leaving:
[tex]\[ -2(x + 1) \][/tex]
5. Expand and Simplify: Finally, distribute the -2 across the \(x + 1\):
[tex]\[ -2(x + 1) = -2x - 2 \][/tex]
Thus, the simplified form of \(\frac{2x^2 - 2}{-x + 1}\) is:
[tex]\[ \boxed{-2x - 2} \][/tex]
So, the correct answer is:
C. [tex]\(-2x - 2\)[/tex]
1. Factor the Numerator: Start by factoring the numerator \(2x^2 - 2\). Notice that both terms in the numerator share a common factor of 2:
[tex]\[ 2x^2 - 2 = 2(x^2 - 1) \][/tex]
2. Recognize a Difference of Squares: Recognize that \(x^2 - 1\) is a difference of squares, which can be factored further:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
Substituting this back into the expression, we have:
[tex]\[ 2(x^2 - 1) = 2(x - 1)(x + 1) \][/tex]
3. Rewrite the Expression: Substitute the factored form into the original expression:
[tex]\[ \frac{2(x - 1)(x + 1)}{-x + 1} \][/tex]
4. Simplify by Canceling Common Factors: Observe that \((x - 1)\) and \((-x + 1)\) are related. Note that \((-x + 1)\) can be rewritten as \(-(x - 1)\). So, the expression becomes:
[tex]\[ \frac{2(x - 1)(x + 1)}{-(x - 1)} = -2(x + 1) \][/tex]
Here, the \((x - 1)\) terms cancel out, leaving:
[tex]\[ -2(x + 1) \][/tex]
5. Expand and Simplify: Finally, distribute the -2 across the \(x + 1\):
[tex]\[ -2(x + 1) = -2x - 2 \][/tex]
Thus, the simplified form of \(\frac{2x^2 - 2}{-x + 1}\) is:
[tex]\[ \boxed{-2x - 2} \][/tex]
So, the correct answer is:
C. [tex]\(-2x - 2\)[/tex]
