Which of the following expressions is equivalent to \(2\mathrm{a}^2(\mathrm{a} + 3)\)?

1. INFER the solution strategy

• Given: \(2\mathrm{a}^2(\mathrm{a} + 3)\)
• Strategy needed: Apply the distributive property to multiply the monomial by each term in the binomial

2. SIMPLIFY using the distributive property

• Distribute \(2\mathrm{a}^2\) to each term inside the parentheses:

\(2\mathrm{a}^2(\mathrm{a} + 3) = 2\mathrm{a}^2 \cdot \mathrm{a} + 2\mathrm{a}^2 \cdot 3\)

3. SIMPLIFY each multiplication

• First term:

\(2\mathrm{a}^2 \cdot \mathrm{a} = 2\mathrm{a}^{(2+1)} = 2\mathrm{a}^3\)

• Second term:

\(2\mathrm{a}^2 \cdot 3 = 6\mathrm{a}^2\)

4. Write the final expanded form

• \(2\mathrm{a}^2(\mathrm{a} + 3) = 2\mathrm{a}^3 + 6\mathrm{a}^2\)

Answer: D. \(2\mathrm{a}^3 + 6\mathrm{a}^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete SIMPLIFY execution: Students distribute \(2\mathrm{a}^2\) to only the first term \((\mathrm{a})\) but forget to distribute it to the second term \((3)\).

They might think: \(2\mathrm{a}^2(\mathrm{a} + 3) = 2\mathrm{a}^3 + 3\), stopping after multiplying by just the first term.

This may lead them to select Choice C \((2\mathrm{a}^3 + 3)\)

Second Most Common Error:

Weak exponent rule knowledge: Students make errors when combining exponents, such as adding coefficients instead of multiplying them, or incorrectly handling the exponent arithmetic.

They might calculate \(2\mathrm{a}^2 \cdot \mathrm{a}\) as \(5\mathrm{a}^3\) (adding 2 + a² + a somehow) or create other exponent errors.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically apply the distributive property while correctly handling exponent rules - both skills must work together for success.