When simplified, which expressions produce the trinomial 4x^2 - 20x - 24? Select the two that apply. ( 2 points)

−4(x^2+5x+6)
4(x^2−20x−24)***********
4(x−6)(x+1)
4(x^2−5x−6)***********

1 answer

To determine which given expressions simplify to the trinomial \( 4x^2 - 20x - 24 \), we'll simplify each expression one by one.

  1. 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 \( 4x^2 - 20x - 24 \).

  2. 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.

  3. 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.

  4. 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:

  • \( 4(x - 6)(x + 1) \)
  • \( 4(x^2 - 5x - 6) \)