8x^2 + 12xWhich of the following is equivalent to the given expression?\(2\mathrm{x}(4\mathrm{x} + 5)\)\(4\mathrm{x}(2\mathrm{x} + 3)\)\(4\mathrm{x}(2...

1. INFER the solution strategy

• The expression \(8\mathrm{x}^2 + 12\mathrm{x}\) needs to be rewritten in factored form
• Strategy: Find the greatest common factor (GCF) of both terms and factor it out
• This will give us an equivalent expression in the form: (common factor) × (remaining expression)

2. SIMPLIFY to find the GCF

• For \(8\mathrm{x}^2\): factors are 8, x, and x
• For \(12\mathrm{x}\): factors are 12 and x
• GCF of coefficients: 8 = \(2^3\) and 12 = \(2^2 \times 3\), so GCF = 4
• GCF of variables: both terms have at least one x
• Overall GCF = \(4\mathrm{x}\)

3. SIMPLIFY by factoring out the GCF

• \(8\mathrm{x}^2 \div 4\mathrm{x} = 2\mathrm{x}\)
• \(12\mathrm{x} \div 4\mathrm{x} = 3\)
• Therefore: \(8\mathrm{x}^2 + 12\mathrm{x} = 4\mathrm{x}(2\mathrm{x} + 3)\)

4. SIMPLIFY to verify the answer

• Expand \(4\mathrm{x}(2\mathrm{x} + 3)\): \(4\mathrm{x}(2\mathrm{x}) + 4\mathrm{x}(3) = 8\mathrm{x}^2 + 12\mathrm{x}\) ✓
• This matches our original expression

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when dividing terms by the GCF, particularly with the constant term.

For example, after correctly identifying \(4\mathrm{x}\) as the GCF, they might calculate:

• \(8\mathrm{x}^2 \div 4\mathrm{x} = 2\mathrm{x}\) ✓ (correct)
• \(12\mathrm{x} \div 4\mathrm{x} = 12\) ✗ (incorrect - should be 3)

This leads them to write \(8\mathrm{x}^2 + 12\mathrm{x} = 4\mathrm{x}(2\mathrm{x} + 12)\), causing them to select Choice C (\(4\mathrm{x}(2\mathrm{x} + 12)\)).

Second Most Common Error:

Poor INFER reasoning about GCF: Students factor out only part of the greatest common factor, typically just the variable portion.

They might factor out only \(2\mathrm{x}\): \(8\mathrm{x}^2 + 12\mathrm{x} = 2\mathrm{x}(4\mathrm{x} + 6)\), then make an additional arithmetic error to get \(2\mathrm{x}(4\mathrm{x} + 5)\), leading them to select Choice A (\(2\mathrm{x}(4\mathrm{x} + 5)\)).

The Bottom Line:

This problem requires careful attention to both finding the complete GCF and executing the division accurately. Students often rush through the arithmetic steps after correctly identifying the strategy.