Which expression represents the completely factored form of 12xy^2 + 8x^2y + 4xy?

1. TRANSLATE the problem requirements

• Given: \(\mathrm{12xy^2 + 8x^2y + 4xy}\)
• Need: Completely factored form
• What this means: Factor out all common factors so no further factoring is possible

2. INFER the factoring strategy

• To factor completely, we must find the Greatest Common Factor (GCF) of all terms
• The GCF includes both coefficient and variable parts
• Strategy: Find GCF of coefficients, then find common variable factors

3. SIMPLIFY to find the GCF of coefficients

• Coefficients are: 12, 8, 4
• Find factors systematically:
  • \(\mathrm{12 = 2^2 \times 3}\)
  • \(\mathrm{8 = 2^3}\)
  • \(\mathrm{4 = 2^2}\)
• \(\mathrm{GCF = 2^2 = 4}\)

4. SIMPLIFY to find common variable factors

• Look at each variable separately:
• For x: appears as \(\mathrm{x^1}\) in \(\mathrm{12xy^2}\), \(\mathrm{x^2}\) in \(\mathrm{8x^2y}\), \(\mathrm{x^1}\) in \(\mathrm{4xy}\)
  • Minimum power = \(\mathrm{x^1 = x}\)
• For y: appears as \(\mathrm{y^2}\) in \(\mathrm{12xy^2}\), \(\mathrm{y^1}\) in \(\mathrm{8x^2y}\), \(\mathrm{y^1}\) in \(\mathrm{4xy}\)
  • Minimum power = \(\mathrm{y^1 = y}\)
• Common variable factor = \(\mathrm{xy}\)

5. SIMPLIFY by factoring out the complete GCF

• Complete GCF = \(\mathrm{4xy}\)
• Divide each term by \(\mathrm{4xy}\):
  • \(\mathrm{12xy^2 \div 4xy = 3y}\)
  • \(\mathrm{8x^2y \div 4xy = 2x}\)
  • \(\mathrm{4xy \div 4xy = 1}\)
• Result: \(\mathrm{4xy(3y + 2x + 1)}\)

6. INFER verification step needed

• Expand \(\mathrm{4xy(3y + 2x + 1)}\) to check:
• \(\mathrm{4xy \times 3y + 4xy \times 2x + 4xy \times 1 = 12xy^2 + 8x^2y + 4xy}\) ✓

Answer: A) \(\mathrm{4xy(3y + 2x + 1)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make errors when finding the GCF of coefficients or miss the minimum power rule for variables. For example, they might incorrectly identify the GCF as \(\mathrm{2xy}\) instead of \(\mathrm{4xy}\), or use \(\mathrm{x^2}\) instead of \(\mathrm{x^1}\) as the common factor.

This leads to incorrect factorizations like \(\mathrm{2xy(6y + 4x + 2)}\) or similar expressions that don't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Incomplete INFER reasoning: Students might correctly find the GCF but fail to factor out the constant term. They might get \(\mathrm{4xy(3y + 2x)}\) and think they're done, not realizing they haven't accounted for the \(\mathrm{4xy}\) term in the original expression.

This may lead them to select Choice D (\(\mathrm{4xy(3y + 2x)}\)).

The Bottom Line:

This problem requires systematic attention to both numerical and variable components of the GCF. Success depends on methodical execution of the factoring process and verification that all terms have been properly accounted for.