Which expression is a factor of 2x^2 + 38x + 10?

1. TRANSLATE the problem information

• Given: The expression \(\mathrm{2x^2 + 38x + 10}\)
• Need to find: Which choice represents a factor of this expression

2. INFER the approach needed

• A factor divides evenly into an expression
• Terms are the individual parts added/subtracted in an expression
• Strategy: Look for common factors first, then check which choice matches

3. INFER what to look for first

• Check if there's a common factor in all terms:
  • \(\mathrm{2x^2}\): divisible by 2
  • \(\mathrm{38x}\): divisible by 2
  • \(\mathrm{10}\): divisible by 2
• The greatest common factor is 2

4. SIMPLIFY by factoring out the common factor

• Factor out 2: \(\mathrm{2x^2 + 38x + 10 = 2(x^2 + 19x + 5)}\)
• The complete factorization shows factors: 2 and \(\mathrm{(x^2 + 19x + 5)}\)

5. INFER which choice matches our factors

• Compare answer choices with our factors:
  • A. 2 ← This matches one of our factors!
  • B. \(\mathrm{5x}\) ← Not a factor
  • C. \(\mathrm{38x}\) ← This is a term, not a factor
  • D. \(\mathrm{2x^2}\) ← This is a term, not a factor

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about factors vs terms: Students confuse the individual terms in an expression with the factors of that expression.

They see \(\mathrm{38x}\) or \(\mathrm{2x^2}\) in the original expression and think "these appear in the expression, so they must be factors." However, terms are the parts that are added or subtracted, while factors are expressions that multiply together to give the original expression.

This may lead them to select Choice C (\(\mathrm{38x}\)) or Choice D (\(\mathrm{2x^2}\)).

Second Most Common Error:

Weak INFER skill: Students don't recognize that they need to look for common factors first, or they don't understand what "factor" means mathematically.

Without a systematic approach to factoring, they might randomly select an answer or get confused about the difference between factoring and just identifying parts of an expression.

This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students understand the fundamental difference between factors (things that multiply to give the expression) and terms (things that are added/subtracted in the expression). Success requires recognizing common factors and applying basic factoring techniques.