To determine which of the given expressions simplifies to an equivalent form of the expression \( \frac{2x^3 - 2x}{x + 1} \), we need to simplify each of the provided options and compare them.
Let's simplify the target expression first:
Simplifying \( \frac{2x^3 - 2x}{x + 1} \):
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Factor out the numerator: \[ 2x^3 - 2x = 2x(x^2 - 1) = 2x(x - 1)(x + 1) \]
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The expression becomes: \[ \frac{2x(x - 1)(x + 1)}{x + 1} \]
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Cancel the \( x + 1 \) terms (assuming \( x \neq -1 \)): \[ 2x(x - 1) \] This simplifies to: \[ 2x^2 - 2x \]
Now let's analyze each option to see if they simplify to \( 2x^2 - 2x \):
Option A: \( \frac{x^2 - 1}{2x^2 + 2x} \)
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Factor the numerator and the denominator: \[ x^2 - 1 = (x - 1)(x + 1) \] \[ 2x^2 + 2x = 2x(x + 1) \]
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The expression becomes: \[ \frac{(x - 1)(x + 1)}{2x(x + 1)} \]
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Cancel the \( x + 1 \) terms (assuming \( x \neq -1 \)): \[ \frac{x - 1}{2x} \]
This does not simplify to \( 2x^2 - 2x \).
Option B: \( \frac{2x^3 - 2x}{x - 1} \)
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This is the original numerator: \[ 2x^3 - 2x = 2x(x^2 - 1) = 2x(x - 1)(x + 1) \]
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The expression becomes: \[ \frac{2x(x - 1)(x + 1)}{x - 1} \]
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Cancel out \( x - 1 \) (assuming \( x \neq 1 \)): \[ 2x(x + 1) \]
This does not simplify to \( 2x^2 - 2x \).
Option C: \( \frac{2x^2 + 6x}{x^2 + 2x - 3} \)
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Factor numerator and denominator: \[ 2x^2 + 6x = 2x(x + 3) \] \[ x^2 + 2x - 3 = (x + 3)(x - 1) \]
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The expression becomes: \[ \frac{2x(x + 3)}{(x + 3)(x - 1)} \]
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Cancel \( x + 3 \) (assuming \( x \neq -3 \)): \[ \frac{2x}{x - 1} \]
This does not simplify to \( 2x^2 - 2x \).
Option D: \( \frac{2x^3 + 2x^2 - 4x}{x + 2} \)
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Factor the numerator: \[ 2x^3 + 2x^2 - 4x = 2x(x^2 + x - 2) = 2x(x + 2)(x - 1) \]
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The expression becomes: \[ \frac{2x(x + 2)(x - 1)}{x + 2} \]
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Cancel \( x + 2 \) (assuming \( x \neq -2 \)): \[ 2x(x - 1) = 2x^2 - 2x \]
This simplifies to \( 2x^2 - 2x \).
Conclusion:
The correct option that simplifies to an equivalent form of \( 2x^2 - 2x \) is:
Option D: \( \frac{2x^3 + 2x^2 - 4x}{x + 2} \).