Which expression shows the complete factorization of 8xy + 12y?\(2y(4x + 6)\)\(4x(2y + 3)\)\(4y(2x + 3)\)\(12y(2x + 1)\)

1. TRANSLATE the problem information

• Given: Find complete factorization of \(\mathrm{8xy + 12y}\)
• What this means: Find the greatest common factor (GCF) and factor it out completely

2. INFER the solution strategy

• To find complete factorization, I need the GCF of all terms
• Complete factorization means no common factors remain in the parentheses
• I'll break each term into prime factors and variables

3. SIMPLIFY to find the GCF

• First term: \(\mathrm{8xy = 2^3 \times x \times y}\)
• Second term: \(\mathrm{12y = 2^2 \times 3 \times y}\)
• GCF of coefficients: 4 (since both contain \(\mathrm{2^2}\))
• GCF of variables: y (since both terms contain y)
• Overall GCF: 4y

4. SIMPLIFY the factorization

• Factor out 4y: \(\mathrm{8xy + 12y = 4y(2x + 3)}\)
• Check: \(\mathrm{4y(2x + 3) = 8xy + 12y}\) ✓

Answer: C. 4y(2x + 3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't understand the difference between factorization and complete factorization. They find a common factor but not the greatest common factor.

For example, they might notice that 2y is a common factor and write: \(\mathrm{8xy + 12y = 2y(4x + 6)}\). While this expands correctly, it's incomplete because 4x + 6 still has a common factor of 2. The complete factorization would be \(\mathrm{2y \times 2(2x + 3) = 4y(2x + 3)}\).

This may lead them to select Choice A (2y(4x + 6))

Second Most Common Error:

Poor SIMPLIFY execution: Students make errors when finding the GCF, particularly with the variable parts. They might incorrectly think x is common to both terms.

For instance, they might write: \(\mathrm{8xy + 12y = 4x(2y + 3)}\), not realizing that the second term (12y) doesn't contain x at all.

This may lead them to select Choice B (4x(2y + 3))

The Bottom Line:

This problem tests whether students truly understand complete factorization versus just any factorization. The key insight is that "complete" means extracting the greatest common factor, leaving no further common factors in the remaining expression.