Consider the expression \(5\mathrm{x}(6\mathrm{x}) + 3\mathrm{x}(2\mathrm{x})\). Which of the following is equivalent to the expression?24x^230x^236x^...

1. INFER the solution approach

• We have two products that need to be computed and then combined
• Strategy: Calculate each monomial product separately, then add the results

2. SIMPLIFY the first product: \(5\mathrm{x}(6\mathrm{x})\)

• Multiply coefficients: \(5 \times 6 = 30\)
• Multiply variables: \(\mathrm{x} \times \mathrm{x} = \mathrm{x}^2\)
• Result: \(5\mathrm{x}(6\mathrm{x}) = 30\mathrm{x}^2\)

3. SIMPLIFY the second product: \(3\mathrm{x}(2\mathrm{x})\)

• Multiply coefficients: \(3 \times 2 = 6\)
• Multiply variables: \(\mathrm{x} \times \mathrm{x} = \mathrm{x}^2\)
• Result: \(3\mathrm{x}(2\mathrm{x}) = 6\mathrm{x}^2\)

4. SIMPLIFY by adding like terms

• Combine: \(30\mathrm{x}^2 + 6\mathrm{x}^2 = 36\mathrm{x}^2\)

Answer: C (\(36\mathrm{x}^2\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students compute only the first term \(5\mathrm{x}(6\mathrm{x}) = 30\mathrm{x}^2\) and forget to include the second term \(3\mathrm{x}(2\mathrm{x})\).

They see "\(5\mathrm{x}(6\mathrm{x}) + 3\mathrm{x}(2\mathrm{x})\)" but focus only on the more prominent first product, losing track of the addition. This incomplete solution leads them to select Choice B (\(30\mathrm{x}^2\)).

Second Most Common Error:

Conceptual confusion about exponent rules: Students correctly multiply the coefficients but fail to apply \(\mathrm{x} \times \mathrm{x} = \mathrm{x}^2\), instead treating the variable multiplication as \(\mathrm{x} + \mathrm{x} = 2\mathrm{x}\) or simply leaving it as x.

For example, they might calculate \(5\mathrm{x}(6\mathrm{x}) = 30\mathrm{x}\) and \(3\mathrm{x}(2\mathrm{x}) = 6\mathrm{x}\), leading to confusion about how to match their result with the \(\mathrm{x}^2\) format of the answer choices. This causes them to get stuck and guess.

The Bottom Line:

This problem tests systematic algebraic manipulation. Students must methodically work through each product and resist the urge to skip steps or lose track of terms in the expression.