Which expression is equivalent to \((\mathrm{x}^2 + 11)^2\) + \((\mathrm{x} - 5)(\mathrm{x} + 5)\)?

1. TRANSLATE the problem information

• Given expression: \((\mathrm{x}^2 + 11)^2 + (\mathrm{x} - 5)(\mathrm{x} + 5)\)
• Need to find: Equivalent simplified expression

2. INFER the most efficient approach

• The first term \((\mathrm{x}^2 + 11)^2\) requires squaring a binomial
• The second term \((\mathrm{x} - 5)(\mathrm{x} + 5)\) is a difference of squares pattern
• Strategy: Expand each term separately, then combine

3. SIMPLIFY the first term: \((\mathrm{x}^2 + 11)^2\)

• Rewrite as: \((\mathrm{x}^2 + 11)(\mathrm{x}^2 + 11)\)
• Use FOIL method:
  • First: \(\mathrm{x}^2 \times \mathrm{x}^2 = \mathrm{x}^4\)
  • Outer: \(\mathrm{x}^2 \times 11 = 11\mathrm{x}^2\)
  • Inner: \(11 \times \mathrm{x}^2 = 11\mathrm{x}^2\)
  • Last: \(11 \times 11 = 121\)
• Result: \(\mathrm{x}^4 + 11\mathrm{x}^2 + 11\mathrm{x}^2 + 121 = \mathrm{x}^4 + 22\mathrm{x}^2 + 121\)

4. INFER the pattern for the second term

• \((\mathrm{x} - 5)(\mathrm{x} + 5)\) follows the difference of squares pattern: \((\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b}) = \mathrm{a}^2 - \mathrm{b}^2\)
• Here, \(\mathrm{a} = \mathrm{x}\) and \(\mathrm{b} = 5\)
• So \((\mathrm{x} - 5)(\mathrm{x} + 5) = \mathrm{x}^2 - 5^2 = \mathrm{x}^2 - 25\)

5. SIMPLIFY by combining both parts

• \((\mathrm{x}^2 + 11)^2 + (\mathrm{x} - 5)(\mathrm{x} + 5) = (\mathrm{x}^4 + 22\mathrm{x}^2 + 121) + (\mathrm{x}^2 - 25)\)
• Combine like terms: \(\mathrm{x}^4 + 22\mathrm{x}^2 + \mathrm{x}^2 + 121 - 25\)
• Final result: \(\mathrm{x}^4 + 23\mathrm{x}^2 + 96\)

Answer: B. \(\mathrm{x}^4 + 23\mathrm{x}^2 + 96\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize \((\mathrm{x} - 5)(\mathrm{x} + 5)\) as a difference of squares pattern and instead use FOIL, leading to calculation errors or taking longer than necessary.

When expanding \((\mathrm{x} - 5)(\mathrm{x} + 5)\) with FOIL, they get \(\mathrm{x}^2 - 5\mathrm{x} + 5\mathrm{x} - 25\), but might make sign errors or forget that \(-5\mathrm{x} + 5\mathrm{x} = 0\), potentially getting confused about the middle term.

This computational confusion may lead them to select Choice A (\(\mathrm{x}^4 + 23\mathrm{x}^2 - 14\)) or cause them to abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when combining like terms, particularly when adding \(22\mathrm{x}^2 + \mathrm{x}^2 = 23\mathrm{x}^2\) or when handling the constants \(121 - 25 = 96\).

A common mistake is getting the coefficient of \(\mathrm{x}^2\) wrong (maybe calculating \(22 + 1 = 24\) instead of 23) or making sign errors with the constants, leading to incorrect final expressions.

This may lead them to select Choice C (\(\mathrm{x}^4 + 12\mathrm{x}^2 + 121\)) if they forgot to add the \(\mathrm{x}^2\) term from the second expression.

The Bottom Line:

This problem tests both pattern recognition and careful algebraic manipulation. Success depends on efficiently recognizing the difference of squares pattern and maintaining accuracy through multiple steps of combining terms.