Which of the following is equivalent to \(2(\mathrm{x}^2 - \mathrm{x}) + 3(\mathrm{x}^2 - \mathrm{x})\)?

1. INFER the structure of the problem

• Given expression: \(2(x^2 - x) + 3(x^2 - x)\)
• Key insight: Both terms contain the identical expression \((x^2 - x)\)
• This means we can treat \((x^2 - x)\) as a common factor

2. INFER the strategy

• Since both terms have the same expression in parentheses, we can combine the coefficients
• Think of this like: \(2(\text{something}) + 3(\text{something}) = 5(\text{something})\)
• Here: \(2(x^2 - x) + 3(x^2 - x) = 5(x^2 - x)\)

3. SIMPLIFY by distributing

• Now distribute the 5: \(5(x^2 - x)\)
• Apply distributive property: \(5(x^2) + 5(-x)\)
• Final result: \(5x^2 - 5x\)

Answer: A. \(5x^2 - 5x\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly get to \(5(x^2 - x)\) but make a sign error when distributing.

They might think: \(5(x^2 - x) = 5x^2 + 5x\), forgetting that the negative sign in front of x needs to be multiplied by 5 as well. The correct distribution is \(5(x^2) + 5(-x) = 5x^2 - 5x\).

This leads them to select Choice B (\(5x^2 + 5x\)).

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the common factor structure and instead try to distribute each term separately first.

They might expand to: \(2x^2 - 2x + 3x^2 - 3x\), then make arithmetic errors when combining, potentially losing entire terms and ending up with incomplete expressions.

This may lead them to select Choice C (\(5x\)) or Choice D (\(5x^2\)).

The Bottom Line:

This problem tests whether students can recognize when expressions can be factored before expanding, which is often more efficient than distributing first. The key insight is seeing \((x^2 - x)\) as a single "thing" that appears twice.