Which expression is equivalent to \((2\mathrm{x} + 3)(4\mathrm{x} - 1)\)?

1. INFER the required approach

• Given: \((2x + 3)(4x - 1)\) needs to be expanded
• Strategy: Use the distributive property systematically through FOIL method
• This means multiply every term in the first binomial by every term in the second binomial

2. SIMPLIFY using FOIL method

• First terms: \(2x \times 4x = 8x²\)
• Outer terms: \(2x \times (-1) = -2x\)
• Inner terms: \(3 \times 4x = 12x\)
• Last terms: \(3 \times (-1) = -3\)

3. SIMPLIFY by combining all terms

• Write out all products: \(8x² - 2x + 12x - 3\)
• Combine like terms: \(-2x + 12x = 10x\)
• Final result: \(8x² + 10x - 3\)

Answer: C (\(8x² + 10x - 3\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly apply FOIL but fail to combine like terms properly.

They get: \(8x² - 2x + 12x - 3\), but then write the middle terms as just "12x" instead of combining \(-2x + 12x = 10x\). This happens when students don't recognize that -2x and +12x are like terms that must be combined.

This may lead them to select Choice B (\(8x² + 12x - 3\))

Second Most Common Error:

Poor SIMPLIFY execution with signs: Students make sign errors when multiplying terms involving negative numbers.

The most common mistake is treating \(3 \times (-1)\) as +3 instead of -3, or getting confused about the sign when multiplying \(2x \times (-1)\). This creates sign confusion throughout their work.

This may lead them to select Choice D (\(8x² + 10x + 3\))

The Bottom Line:

Success requires systematic application of FOIL combined with careful attention to signs and proper combining of like terms. Students who rush through the combining step or aren't careful with negative signs will select incorrect answers even if they understand the FOIL concept.