To determine which given expressions simplify to the trinomial \( 4x^2 - 20x - 24 \), we'll simplify each expression one by one.
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Expression: \( -4(x^2 + 5x + 6) \)
\[ -4(x^2 + 5x + 6) = -4x^2 - 20x - 24 \]
This simplifies to \( -4x^2 - 20x - 24 \), which does not match \( 4x^2 - 20x - 24 \).
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Expression: \( 4(x^2 - 20x - 24) \)
\[ 4(x^2 - 20x - 24) = 4x^2 - 80x - 96 \]
This simplifies to \( 4x^2 - 80x - 96 \), which does not match.
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Expression: \( 4(x - 6)(x + 1) \)
First, we expand \( (x - 6)(x + 1) \): \[ (x - 6)(x + 1) = x^2 - 6x + x - 6 = x^2 - 5x - 6 \] Now multiplying by 4: \[ 4(x^2 - 5x - 6) = 4x^2 - 20x - 24 \]
This simplifies to \( 4x^2 - 20x - 24 \), which matches.
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Expression: \( 4(x^2 - 5x - 6) \)
As we found just above: \[ 4(x^2 - 5x - 6) = 4x^2 - 20x - 24 \]
This also simplifies to \( 4x^2 - 20x - 24 \), which matches.
Based on our simplifications, the two expressions that produce the trinomial \( 4x^2 - 20x - 24 \) are:
- \( 4(x - 6)(x + 1) \)
- \( 4(x^2 - 5x - 6) \)
Thus, the correct selections are:
- \( 4(x - 6)(x + 1) \)
- \( 4(x^2 - 5x - 6) \)