Which expression is equivalent to \((\mathrm{x}^2 + 5)^2 + (\mathrm{x}^2 - 4)(\mathrm{x}^2 + 4)\)?2x^4 + 10x^2 - 92x^4 + 10x^2...

1. INFER the most efficient expansion strategy

• Given: \((\mathrm{x}^2 + 5)^2 + (\mathrm{x}^2 - 4)(\mathrm{x}^2 + 4)\)
• Key insight: The first term is a perfect square pattern, the second term is a difference of squares pattern
• Strategy: Use special formulas rather than expanding term by term

2. SIMPLIFY the first term using perfect squares

• \((\mathrm{x}^2 + 5)^2\) follows the pattern \((\mathrm{a} + \mathrm{b})^2 = \mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\)
• Where \(\mathrm{a} = \mathrm{x}^2\) and \(\mathrm{b} = 5\):
  • \(\mathrm{a}^2 = (\mathrm{x}^2)^2 = \mathrm{x}^4\)
  • \(2\mathrm{ab} = 2(\mathrm{x}^2)(5) = 10\mathrm{x}^2\)
  • \(\mathrm{b}^2 = 5^2 = 25\)
• Result: \(\mathrm{x}^4 + 10\mathrm{x}^2 + 25\)

3. SIMPLIFY the second term using difference of squares

• \((\mathrm{x}^2 - 4)(\mathrm{x}^2 + 4)\) follows the pattern \((\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b}) = \mathrm{a}^2 - \mathrm{b}^2\)
• Where \(\mathrm{a} = \mathrm{x}^2\) and \(\mathrm{b} = 4\):
  • \(\mathrm{a}^2 = (\mathrm{x}^2)^2 = \mathrm{x}^4\)
  • \(\mathrm{b}^2 = 4^2 = 16\)
• Result: \(\mathrm{x}^4 - 16\)

4. SIMPLIFY by combining the expressions

• \((\mathrm{x}^4 + 10\mathrm{x}^2 + 25) + (\mathrm{x}^4 - 16)\)
• Combine like terms:
  • \(\mathrm{x}^4\) terms: \(\mathrm{x}^4 + \mathrm{x}^4 = 2\mathrm{x}^4\)
  • \(\mathrm{x}^2\) terms: \(10\mathrm{x}^2\) (only one)
  • Constants: \(25 + (-16) = 9\)
• Final result: \(2\mathrm{x}^4 + 10\mathrm{x}^2 + 9\)

Answer: B. \(2\mathrm{x}^4 + 10\mathrm{x}^2 + 9\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the special algebraic patterns and try to expand everything using FOIL or distribution, leading to more complex calculations and increased chance of arithmetic errors.

For example, they might expand \((\mathrm{x}^2 + 5)^2\) as \((\mathrm{x}^2 + 5)(\mathrm{x}^2 + 5)\) and use FOIL: \(\mathrm{x}^2 \cdot \mathrm{x}^2 + \mathrm{x}^2 \cdot 5 + 5 \cdot \mathrm{x}^2 + 5 \cdot 5\). While this gives the same result, it's more error-prone and time-consuming.

This approach increases the likelihood of arithmetic mistakes that could lead them to select Choice A (\(2\mathrm{x}^4 + 10\mathrm{x}^2 - 9\)) or other incorrect options.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the patterns but make sign errors when combining terms, particularly with the constant terms (\(25 - 16 = 9\)).

A common mistake is getting confused with the signs and calculating \(25 + 16 = 41\) or \(25 - 16 = -9\), leading them to select Choice A (\(2\mathrm{x}^4 + 10\mathrm{x}^2 - 9\)) when they get -9 instead of +9.

The Bottom Line:

This problem rewards students who can quickly recognize standard algebraic patterns and apply the corresponding formulas efficiently, while penalizing those who take longer, more error-prone approaches or make arithmetic mistakes in the final combining step.