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Given the expression 2x2+4x2x2−8⋅x2−2x+1x2−1%0D%0A2%0D%0A𝑥%0D%0A2%0D%0A+%0D%0A4%0D%0A𝑥%0D%0A2%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A8%0D%0A⋅%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A2%0D%0A𝑥%0D%0A+%0D%0A1%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A1%0D%0A , which option is the correct simplification?%0D%0A%0D%0AOption #1: 2x4(x+2)%0D%0A2%0D%0A𝑥%0D%0A4%0D%0A(%0D%0A𝑥%0D%0A+%0D%0A2%0D%0A)%0D%0A %0D%0A%0D%0AOption #2: xx−2%0D%0A𝑥%0D%0A𝑥%0D%0A−%0D%0A2%0D%0A %0D%0A%0D%0AOption #3: x(x−1)(x−2)(x+1)

1 answer

To simplify the expression \( 2x^2 + 4x^2x^2 - 8 \cdot x^2 - 2x + 1x^2 - 1 \), let's first rewrite it more clearly:

  1. Combine like terms:

    • The expression can be rewritten as: \[ 2x^2 + 4x^4 - 8x^2 - 2x + x^2 - 1 \]
    • Now group the like terms together: \[ (4x^4) + (2x^2 - 8x^2 + x^2) - 2x - 1 \] \[ 4x^4 - 5x^2 - 2x - 1 \]

  2. Attempt to factor this expression:

    • At first glance, a quartic polynomial like \( 4x^4 - 5x^2 - 2x - 1 \) may not factor easily, but we can try to look for rational roots or try grouping.
    • The polynomial can be tested for possible factorizations, or we can use synthetic division or the Rational Root Theorem.

  3. Solving for roots:

    • Let's look for rational roots \( x = -1 \) and \( x = 1 \).
    • Testing these might reveal the factors.

  4. Finding factors:

    • If one solution is \( x = 1 \), we can factor by: \[ 4x^4 - 5x^2 - 2x - 1 = (x - 1)(4x^3 + 4x^2 - x - 1) \]

Now, we want to compare this with the given options:

  • Option #1: \( 2x^4(x + 2) \)

    • This does not match as the terms do not align.

  • Option #2: \( x(x - 2) \)

    • This simplifies to \( x^2 - 2x \), which does not represent the quartic.

  • Option #3: \( x(x - 1)(x - 2)(x + 1) \)

    • While this might look promising, we have a quartic equation an not cubic or lower degree.

This means that the expressions do not immediately simplify into the options given. The first two options clearly do not factor out correctly, and the third option is similarly not a direct match.

Final Verification: Upon attempting to rearrange or recognizing that none of these perfectly represent the quartic or its roots.

Conclusively, it appears there might be a mistake or need for a deeper analysis with no given option being valid based directly on initial simplifications. Hence, the correct response cannot be derived correctly from the options provided without conflicting interpretations.

Please check options for clarification; otherwise, further simplifications might be needed.