Which expression is equivalent to \((\mathrm{x}^2 + 3)^2 - (\mathrm{x} - 2)(\mathrm{x} + 2)\)?

1. INFER the algebraic patterns present

• Given expression: \((x^2 + 3)^2 - (x - 2)(x + 2)\)
• Recognize two key patterns:
  • \((x^2 + 3)^2\) is a perfect square
  • \((x - 2)(x + 2)\) is a difference of squares

2. SIMPLIFY the perfect square term

• Apply \((a + b)^2 = a^2 + 2ab + b^2\) with \(a = x^2\) and \(b = 3\):

\((x^2 + 3)^2 = (x^2)^2 + 2(x^2)(3) + 3^2\)
\(= x^4 + 6x^2 + 9\)

3. SIMPLIFY the difference of squares term

• Apply \((a - b)(a + b) = a^2 - b^2\) with \(a = x\) and \(b = 2\):

\((x - 2)(x + 2) = x^2 - 2^2\)
\(= x^2 - 4\)

4. SIMPLIFY by combining the expressions

• Substitute back into original expression:

\(x^4 + 6x^2 + 9 - (x^2 - 4)\)

• Distribute the negative sign carefully:

\(x^4 + 6x^2 + 9 - x^2 + 4\)

• Combine like terms:

\(x^4 + (6x^2 - x^2) + (9 + 4)\)
\(= x^4 + 5x^2 + 13\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign error when subtracting \((x^2 - 4)\)

Many students correctly expand both parts but write:

\(x^4 + 6x^2 + 9 - (x^2 - 4) = x^4 + 6x^2 + 9 - x^2 - 4 = x^4 + 5x^2 + 5\)

They forget that subtracting \((x^2 - 4)\) means subtracting both \(x^2\) AND \(-4\), so they should get \(+4\), not \(-4\).

This may lead them to select Choice C \((x^4 + 6x^2 + 5)\) - though this specific error doesn't match any given choice, leading to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge: Not recognizing the difference of squares pattern

Students might try to expand \((x - 2)(x + 2)\) using FOIL instead of the shortcut formula:

\((x - 2)(x + 2) = x^2 + 2x - 2x - 4 = x^2 - 4\)

While this gives the right answer, the extra steps create more opportunities for arithmetic errors, and some students make mistakes in the FOIL process itself.

This leads to confusion and potentially guessing among the answer choices.

The Bottom Line:

This problem tests your ability to recognize algebraic patterns and execute multi-step simplification without making sign errors. The key insight is seeing that both expressions have special forms that make expansion more efficient.