To find the expressions equivalent to \( 2x^2 - 8x - 10 \) when distributed, we need to check which of the provided options expand to this polynomial.
Option #1: \( 2x(x - 4x - 5) \)
This seems to be incorrectly expressed; however, it looks like it was intended to be \( 2x(x - 4) - 10 \).
Distributing gives:
\[ 2x(x - 4 - 5) = 2x(-5) = -10x \]
This is not equivalent.
Option #2: \( (2x + 2)(x - 5) \)
Distributing gives:
\[ 2x \cdot x + 2 \cdot x - 2x \cdot 5 - 2 \cdot 5 = 2x^2 + 2x - 10x - 10 \]
\[ = 2x^2 - 8x - 10 \]
This is equivalent!
Option #3: \( 2x(x - 5) + 2(x - 5) \)
Factoring out \( (x - 5) \):
\[ (2x + 2)(x - 5) \]
This is the same as Option #2, and we know this is equivalent.
Option #4: \( (x - 5)(x - 3) \)
Distributing gives:
\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]
This is not equivalent.
Option #5: \(-6x - 10\)
This is a linear expression and does not match the quadratic form.
Thus the 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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