Which of the following is equivalent to 6ab + 9a?\(\mathrm{3a(2b + 3a)}\)\(\mathrm{3a(2b + 3)}\)\(\mathrm{a(6b + 3)}\)\(\mathrm{9a(2b + 1)}\)

1. INFER the solution approach

• This expression has two terms: \(6ab + 9a\)
• Both terms share common factors, so I need to factor out the Greatest Common Factor (GCF)
• Strategy: Find GCF of both terms, then factor it out

2. INFER what the GCF contains

• Look at numerical coefficients: \(6\) and \(9\)
  • Factors of \(6: 1, 2, 3, 6\)
  • Factors of \(9: 1, 3, 9\)
  • Greatest common factor of coefficients: \(3\)
• Look at variables: both terms contain \(a\), but only first term has \(b\)
  • Common variable factor: \(a\)
• Overall GCF = \(3a\)

3. SIMPLIFY by factoring out the GCF

• \(6ab + 9a = 3a(2b) + 3a(3)\)
• Factor out \(3a\): = \(3a(2b + 3)\)

4. SIMPLIFY through verification

• Expand to check: \(3a(2b + 3) = 6ab + 9a\) ✓
• This matches our original expression

Answer: B. \(3a(2b + 3)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't systematically find the complete GCF, missing either the numerical factor or variable factor.

For example, they might only factor out \(a\) and get \(a(6b + 9)\), which isn't simplified completely. Or they might factor out \(3\) and get \(3(2ab + 3a)\), which still has common factors remaining. This incomplete factoring leads to confusion when none of the answer choices match their work, causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need to factor out \(3a\), but make errors in the algebra.

They might incorrectly think \(6ab \div 3a = 2ab\) instead of \(2b\), leading them to \(3a(2ab + 3)\). This would make them select Choice A \((3a(2b + 3a))\) since it's the closest to their incorrect work.

The Bottom Line:

The key challenge is being systematic about finding the complete GCF - both numerical and variable factors - then executing the division carefully when factoring.