To identify which options are equivalent to the expression \( 2x^2 - 8x - 10 \) when distributed, we need to simplify each option and compare them to the original expression.
Original expression:
\[ 2x^2 - 8x - 10 \]
Analyzing the options:
Option #1: \( 2x(x - 4x - 5) \)
This option appears to have a mistake in the parentheses because it has two terms subtracted after \( 4x \). To clarify, it seems intended to be \( 2x(x - 4) - 5 \), but it requires correct interpretation. Simplifying \( 2x(x - 4 - 5) \) gives \( 2x(-5) = -10x \), which is incorrect.
Option #2: \( (2x + 2)(x - 5) \)
Distributing: \[ (2x + 2)(x - 5) = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10 \] This is equivalent to the original expression.
Option #3: \( 2x(x - 5) + 2(x - 5) \)
Factoring out \( (x - 5) \): \[ 2x(x - 5) + 2(x - 5) = (2x + 2)(x - 5) \] From Option #2, this is identified as equivalent to: \[ 2x^2 - 8x - 10 \]
Option #4: \( (x - 5)(x - 3) \)
Distributing gives: \[ (x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \] This is not equivalent to the original expression.
Option #5: \( -6x - 10 \)
This is also not equivalent to the original expression.
Conclusion:
The two equivalent expressions are:
- Option #2: \( (2x + 2)(x - 5) \)
- Option #3: \( 2x(x - 5) + 2(x - 5) \)
So the answer is: Option #2 and Option #3
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