To determine which options are equivalent to the expression \(2x^2 - 8x - 10\) when distributed, we need to simplify each option and see if it matches the original expression.
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Option #1: \(2x(x - 4x - 5)\)
This expression has a mistake in parentheses, it should be \(2x(x - 4)\) since it's not a standard syntax. Let's consider it as \(2x(x - 4 - 5)\) which gives \(2x(x - 9)\).
Distributed:
\[ 2x \cdot x - 18x = 2x^2 - 18x \]
This does not match \(2x^2 - 8x - 10\). -
Option #2: \((2x + 2)(x - 5)\)
Distribution gives:
\[ 2x \cdot x + 2 \cdot x - 10x - 10 = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10 \]
This matches! -
Option #3: \(2x(x - 5) + 2(x - 5)\)
This can be factored as:
\[ (2x + 2)(x - 5) \]
We already determined this as Option #2. -
Option #4: \((x - 5)(x - 3)\)
Distributing yields:
\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
This does not match \(2x^2 - 8x - 10\). -
Option #5: \(-6x - 10\)
This is clearly not equivalent to \(2x^2 - 8x - 10\).
Thus, the two equivalent options from the given list are Option #2 and Option #3.
The final answer would be: Option #2 and Option #3.