Question:Let \(\mathrm{P(x) = (x^2 - 4)^2 - 3(x - 6)(x + 6)}\). Which expression is equivalent to \(\mathrm{P(x)}\)?x^4 - 5x^2...

1. INFER which algebraic patterns to recognize

• Given: \(\mathrm{P(x) = (x^2 - 4)^2 - 3(x - 6)(x + 6)}\)
• Key insight: This has two distinct patterns:
  • \(\mathrm{(x^2 - 4)^2}\) is a perfect square binomial
  • \(\mathrm{(x - 6)(x + 6)}\) is a difference of squares pattern

2. SIMPLIFY the first term using perfect square formula

• Expand \(\mathrm{(x^2 - 4)^2}\):
• Using \(\mathrm{(a - b)^2 = a^2 - 2ab + b^2}\) where \(\mathrm{a = x^2}\) and \(\mathrm{b = 4}\):
• \(\mathrm{(x^2 - 4)^2 = (x^2)^2 - 2(x^2)(4) + 4^2}\)
  \(\mathrm{= x^4 - 8x^2 + 16}\)

3. SIMPLIFY the second term using difference of squares

• Expand \(\mathrm{(x - 6)(x + 6)}\):
• Using \(\mathrm{(a - b)(a + b) = a^2 - b^2}\) where \(\mathrm{a = x}\) and \(\mathrm{b = 6}\):
• \(\mathrm{(x - 6)(x + 6) = x^2 - 36}\)

4. SIMPLIFY by substituting back and distributing

• \(\mathrm{P(x) = (x^4 - 8x^2 + 16) - 3(x^2 - 36)}\)
• Distribute the -3: \(\mathrm{P(x) = x^4 - 8x^2 + 16 - 3x^2 + 108}\)

5. SIMPLIFY by combining like terms

• Group like terms: \(\mathrm{P(x) = x^4 + (-8x^2 - 3x^2) + (16 + 108)}\)
• \(\mathrm{P(x) = x^4 - 11x^2 + 124}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when distributing the negative sign or combining coefficients.

For example, they might calculate \(\mathrm{-8x^2 - 3x^2 = -5x^2}\) instead of \(\mathrm{-11x^2}\), or forget to distribute the -3 properly, getting \(\mathrm{-3x^2 - 108}\) instead of \(\mathrm{-3x^2 + 108}\). This leads them to select Choice A (\(\mathrm{x^4 - 5x^2 - 92}\)) or get confused with the constant term.

Second Most Common Error:

Missing conceptual knowledge: Students don't recognize the algebraic patterns and try to expand everything manually through FOIL.

When expanding \(\mathrm{(x^2 - 4)^2}\) as \(\mathrm{(x^2 - 4)(x^2 - 4)}\), they may make errors in the middle terms or lose track of signs during the lengthy multiplication process. This often leads to incorrect coefficients and causes them to guess among the remaining choices.

The Bottom Line:

This problem tests whether students can efficiently recognize and apply algebraic identities rather than relying on brute-force expansion. Success depends on pattern recognition combined with careful arithmetic.