Which expression is equivalent to \((2\mathrm{x}^2 + 1)(2\mathrm{x} + 3)\)?

1. TRANSLATE the problem information

• We need to find an expression equivalent to \((2\mathrm{x}^2 + 1)(2\mathrm{x} + 3)\)
• This means we need to expand/multiply these two polynomials

2. INFER the solution approach

• This is a polynomial multiplication problem
• We need to use the distributive property: multiply each term in the first polynomial by each term in the second polynomial
• We have two terms in the first polynomial (\(2\mathrm{x}^2\) and \(1\)) and two terms in the second polynomial (\(2\mathrm{x}\) and \(3\))

3. SIMPLIFY by applying the distributive property

• Multiply each term in \((2\mathrm{x}^2 + 1)\) by each term in \((2\mathrm{x} + 3)\):
  • \(2\mathrm{x}^2 \times 2\mathrm{x} = 4\mathrm{x}^3\)
  • \(2\mathrm{x}^2 \times 3 = 6\mathrm{x}^2\)
  • \(1 \times 2\mathrm{x} = 2\mathrm{x}\)
  • \(1 \times 3 = 3\)

4. SIMPLIFY by combining all terms

• \(4\mathrm{x}^3 + 6\mathrm{x}^2 + 2\mathrm{x} + 3\)
• All terms are already in standard polynomial form (highest degree first)

Answer: B. \(4\mathrm{x}^3 + 6\mathrm{x}^2 + 2\mathrm{x} + 3\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors during the multiplication process, particularly when handling positive terms that should remain positive.

For example, they might incorrectly compute \(2\mathrm{x}^2 \times 3 = -6\mathrm{x}^2\) instead of \(+6\mathrm{x}^2\), leading them to select Choice A (\(4\mathrm{x}^3 - 6\mathrm{x}^2 + 2\mathrm{x} + 3\)). Or they might make an error with \(1 \times 2\mathrm{x} = -2\mathrm{x}\) instead of \(+2\mathrm{x}\), leading to Choice C (\(4\mathrm{x}^3 + 6\mathrm{x}^2 - 2\mathrm{x} + 3\)).

Second Most Common Error:

Insufficient INFER reasoning: Students may not systematically apply the distributive property to all term combinations, missing one of the four required multiplications.

This incomplete approach causes confusion about which terms should appear in the final answer, leading them to guess among the choices or select an answer that looks "close enough."

The Bottom Line:

This problem tests careful execution of a fundamental algebraic process. Success requires systematic application of the distributive property with attention to signs - it's not about complex strategy, but about methodical, accurate computation.