A rectangle has a length represented by the expression x^2 - 4x + 2 and a width represented by the...

1. TRANSLATE the problem information

• Given information:
  • Rectangle length: \(\mathrm{x^2 - 4x + 2}\)
  • Rectangle width: \(\mathrm{3x + 5}\)
  • Need to find: Area expression

• What this tells us: Use the rectangle area formula with these polynomial expressions

2. TRANSLATE the geometric concept to algebra

• Rectangle area formula: \(\mathrm{Area = length \times width}\)
• This becomes: \(\mathrm{Area = (x^2 - 4x + 2)(3x + 5)}\)
• Now we have a polynomial multiplication problem

3. SIMPLIFY by multiplying the polynomials

• Use the distributive property - multiply each term in the first polynomial by each term in the second:
  • \(\mathrm{x^2(3x + 5) = 3x^3 + 5x^2}\)
  • \(\mathrm{-4x(3x + 5) = -12x^2 - 20x}\)
  • \(\mathrm{2(3x + 5) = 6x + 10}\)

• This gives us: \(\mathrm{3x^3 + 5x^2 - 12x^2 - 20x + 6x + 10}\)

4. SIMPLIFY by combining like terms

• Group terms by degree:
  • \(\mathrm{x^3}\) terms: \(\mathrm{3x^3}\)
  • \(\mathrm{x^2}\) terms: \(\mathrm{5x^2 - 12x^2 = -7x^2}\)
  • \(\mathrm{x}\) terms: \(\mathrm{-20x + 6x = -14x}\)
  • Constant terms: \(\mathrm{10}\)

Answer: B. \(\mathrm{3x^3 - 7x^2 - 14x + 10}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students make sign errors when distributing negative terms or arithmetic mistakes when combining like terms.

For example, they might incorrectly calculate \(\mathrm{-4x(5)}\) as \(\mathrm{+20x}\) instead of \(\mathrm{-20x}\), or combine \(\mathrm{-20x + 6x}\) as \(\mathrm{-26x}\) instead of \(\mathrm{-14x}\). These calculation errors lead to wrong coefficients in the final expression.

This may lead them to select Choice A (\(\mathrm{3x^3 - 17x^2 - 14x + 10}\)) or Choice C (\(\mathrm{3x^3 - 7x^2 + 26x + 10}\)).

Second Most Common Error:

Weak TRANSLATE reasoning: Students might try to add the length and width expressions instead of multiplying them, forgetting that area requires multiplication.

This conceptual error leads them away from the correct approach entirely, causing confusion and potentially random guessing among the answer choices.

The Bottom Line:

This problem tests whether students can correctly apply the area formula in an algebraic context and execute polynomial multiplication without calculation errors. The multiple similar-looking answer choices are designed to catch common arithmetic mistakes during the simplification process.