Which of the following expressions is a factor of \((2\mathrm{x} - 5)^2 - (\mathrm{x} + 3)^2\)?x + 83x - 2Choose...

1. INFER the pattern in the expression

• Given: \((2x - 5)^2 - (x + 3)^2\)
• This is in the form \(a^2 - b^2\) (difference of squares)
• Where \(a = 2x - 5\) and \(b = x + 3\)

2. INFER the appropriate strategy

• Use the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\)
• This is much faster than expanding both squares first

3. SIMPLIFY to find \((a-b)\)

• \(a - b = (2x - 5) - (x + 3)\)
• \(= 2x - 5 - x - 3\)
• \(= x - 8\)

4. SIMPLIFY to find \((a+b)\)

• \(a + b = (2x - 5) + (x + 3)\)
• \(= 2x - 5 + x + 3\)
• \(= 3x - 2\)

5. Write the factored form and check options

• \((2x - 5)^2 - (x + 3)^2 = (x - 8)(3x - 2)\)
• Option I: \(x + 8\) ✗ (We have \(x - 8\), not \(x + 8\))
• Option II: \(3x - 2\) ✓ (This is exactly one of our factors)

Answer: B (II only)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the difference of squares pattern and instead expand both squared terms first. While this method works, it's longer and creates more opportunities for arithmetic errors during the expansion and subsequent factoring steps. Students may make calculation mistakes when expanding \((2x-5)^2\) or \((x+3)^2\), leading to an incorrect quadratic expression that they then cannot factor properly.

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

Second Most Common Error:

Poor SIMPLIFY execution: Even when students recognize the difference of squares pattern, they make sign errors when computing \((a-b)\). Specifically, they might calculate \((2x-5)-(x+3)\) incorrectly as \(2x-5-x+3 = x-2\) instead of \(x-8\). This gives them factors of \((x-2)(3x-2)\), making them think \(x-2\) might be related to option I.

This may lead them to select Choice A (I only) after incorrectly concluding that \(x+8\) could be a factor.

The Bottom Line:

This problem tests whether students can efficiently recognize and apply the difference of squares formula, avoiding the longer expansion route while maintaining accuracy in sign handling during algebraic manipulations.