To identify which options are equivalent to the expression \(2x^2 - 8x - 10\), let's first factor the original expression, if possible, and then distribute the expressions provided in the options.
Starting with \(2x^2 - 8x - 10\):
- Factoring: First, factor out the common factor of 2: \[ 2(x^2 - 4x - 5) \] Now, let's factor \(x^2 - 4x - 5\): We are looking for two numbers that multiply to \(-5\) and add to \(-4\), which are \(-5\) and \(1\). Therefore, \[ x^2 - 4x - 5 = (x - 5)(x + 1) \] So, \[ 2(x - 5)(x + 1) \]
Now we have \(2x^2 - 8x - 10 = 2(x - 5)(x + 1)\).
Now, let's check each option:
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Option #1: \(2x(x - 4x - 5)\) This option seems incorrect (the term "-4x" is off).
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Option #2: \((2x + 2)(x - 5)\) Distributing: \[ 2x \cdot x + 2 \cdot x - 5 \cdot 2x - 5 \cdot 2 = 2x^2 + 2x - 10x - 10 = 2x^2 - 8x - 10 \] This option is equivalent.
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Option #3: \(2x(x - 5) + 2(x - 5)\) Factoring out \(2(x - 5)\): \[ 2(x - 5)(x + 1) = 2x^2 - 8x - 10 \] This option is also equivalent.
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Option #4: \((x - 5)(x - 3)\) Distributing: \[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \] This is not equivalent.
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Option #5: \(-6x - 10\) This clearly is not equivalent.
Now, the two equivalent options are Option #2 and Option #3.
To arrange them by the lesser number first:
The answer is:
- Option #2: \((2x + 2)(x - 5)\)
- Option #3: \(2x(x - 5) + 2(x - 5)\)