To find two expressions that will be equivalent to \( 2x^2 - 8x - 10 \) when distributed, we will need to distribute each of the options provided and see which match the original expression.
Let's analyze each option:
Option #1: \( 2x(x - 4x - 5) \)
This expression is incorrect in its format, as it has \( -4x \) in parentheses which should not be the case. Let's ignore this option.
Option #2: \( (2x + 2)(x - 5) \)
Distributing: \[ (2x + 2)(x - 5) = 2x(x) + 2(-5) + 2(x) + 2(-5) = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10 \] This matches our original expression.
Option #3: \( 2x(x - 5) + 2(x - 5) \)
Distributing: \[ 2x(x - 5) + 2(x - 5) = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10 \] This also matches our original expression.
Option #4: \( (x - 5)(x - 3) \)
Distributing: \[ (x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \] This does not match the original expression.
Option #5: \( -6x - 10 \)
This expression does not contain any quadratic term and thus does not match the original expression.
Conclusion
The options that match \( 2x^2 - 8x - 10 \) are:
- Option #2: \( (2x + 2)(x - 5) \)
- Option #3: \( 2x(x - 5) + 2(x - 5) \)
Arranging them in order of the lesser number first, we have:
- Option #2: (2x + 2)(x - 5)
- Option #3: 2x(x - 5) + 2(x - 5)
So the final answer is:
- Option #2 (lesser number)
- Option #3