Which expression is equivalent to \((\mathrm{x}+2)^2 - 49\)?

1. INFER the pattern in the expression

• Given: \((\mathrm{x}+2)^2 - 49\)
• Key insight: This is a difference of squares in the form \(\mathrm{a}^2 - \mathrm{b}^2\)
• Here: \(\mathrm{a} = (\mathrm{x}+2)\) and \(\mathrm{b} = 7\) (since \(49 = 7^2\))

2. SIMPLIFY using the difference of squares formula

• Apply the formula: \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a}-\mathrm{b})(\mathrm{a}+\mathrm{b})\)
• Substitute: \((\mathrm{x}+2)^2 - 49 = ((\mathrm{x}+2) - 7)((\mathrm{x}+2) + 7)\)

3. SIMPLIFY the expressions in each factor

• First factor: \((\mathrm{x}+2) - 7 = \mathrm{x} + 2 - 7 = \mathrm{x} - 5\)
• Second factor: \((\mathrm{x}+2) + 7 = \mathrm{x} + 2 + 7 = \mathrm{x} + 9\)
• Final result: \((\mathrm{x} - 5)(\mathrm{x} + 9)\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the difference of squares pattern

Students see \((\mathrm{x}+2)^2 - 49\) and immediately try to expand \((\mathrm{x}+2)^2\) first, getting \(\mathrm{x}^2 + 4\mathrm{x} + 4 - 49 = \mathrm{x}^2 + 4\mathrm{x} - 45\). Then they struggle to factor this trinomial or make sign errors when finding factors of -45 that add to +4. Some students might find factors like -9 and +5 but arrange them incorrectly as \((\mathrm{x}+5)(\mathrm{x}-9)\).

This may lead them to select Choice A (\((\mathrm{x}+5)(\mathrm{x}-9)\)).

Second Most Common Error:

Poor SIMPLIFY execution: Algebraic errors when combining like terms

Students correctly identify the difference of squares pattern but make mistakes when simplifying \((\mathrm{x}+2) \pm 7\). Common errors include:

• \((\mathrm{x}+2) - 7 = \mathrm{x} - 5\) ✓ but \((\mathrm{x}+2) + 7 = \mathrm{x} + 5\) ✗ (forgetting to add \(2+7=9\))
• Sign confusion leading to factors like \((\mathrm{x}+7)(\mathrm{x}-7)\)

This may lead them to select Choice C (\((\mathrm{x}-7)(\mathrm{x}+7)\)).

The Bottom Line:

This problem tests pattern recognition first, algebraic manipulation second. Students who miss the difference of squares shortcut often get bogged down in unnecessary trinomial factoring and make computational errors along the way.