To determine which expressions produce the trinomial \( 4x^2 - 20x - 24 \), we need to simplify each option and see if they match the original expression.
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Option 1: \( -4(x^2 + 5x + 6) \)
- Distributing the -4: \[ -4(x^2 + 5x + 6) = -4x^2 - 20x - 24 \]
- This does not match \( 4x^2 - 20x - 24 \).
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Option 2: \( 4(x - 6)(x + 1) \)
- First we expand \( (x - 6)(x + 1) \): \[ (x - 6)(x + 1) = x^2 + x - 6x - 6 = x^2 - 5x - 6 \]
- Now distributing the 4: \[ 4(x^2 - 5x - 6) = 4x^2 - 20x - 24 \]
- This matches our original trinomial.
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Option 3: \( 4(x^2 - 20x - 24) \)
- This simply multiplies the entire expression by 4: \[ 4(x^2 - 20x - 24) = 4x^2 - 80x - 96 \]
- This does not match the original expression.
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Option 4: \( 4(x^2 - 5x - 6) \)
- As shown in step 2, expanding this expression gives: \[ 4(x^2 - 5x - 6) = 4x^2 - 20x - 24 \]
- This also matches the original trinomial.
Thus, the two expressions that produce \( 4x^2 - 20x - 24 \) are:
- \( 4(x - 6)(x + 1) \)
- \( 4(x^2 - 5x - 6) \)