Question:The length of a rectangle is given by the expression 6x^2 + 3. The width of the rectangle is given...

1. TRANSLATE the problem information

• Given information:
  • Length of rectangle: \(6\mathrm{x}^2 + 3\)
  • Width of rectangle: \(4\mathrm{x}^2 - 1\)
  • Need to find: perimeter expression

• What this tells us: We need to use the rectangle perimeter formula with these algebraic expressions

2. INFER the approach

• Since we have length and width expressions, we need the perimeter formula for rectangles
• Perimeter of rectangle = 2 × length + 2 × width, or \(\mathrm{P} = 2\mathrm{l} + 2\mathrm{w}\)
• We'll substitute our expressions and then simplify

3. Set up the perimeter equation

\(\mathrm{P} = 2\mathrm{l} + 2\mathrm{w}\)
\(\mathrm{P} = 2(6\mathrm{x}^2 + 3) + 2(4\mathrm{x}^2 - 1)\)

4. SIMPLIFY using distributive property

• Apply distributive property to each term:
  • \(2(6\mathrm{x}^2 + 3) = 2(6\mathrm{x}^2) + 2(3) = 12\mathrm{x}^2 + 6\)
  • \(2(4\mathrm{x}^2 - 1) = 2(4\mathrm{x}^2) + 2(-1) = 8\mathrm{x}^2 - 2\)

• So: \(\mathrm{P} = 12\mathrm{x}^2 + 6 + 8\mathrm{x}^2 - 2\)

5. SIMPLIFY by combining like terms

• Group the x² terms: \(12\mathrm{x}^2 + 8\mathrm{x}^2 = 20\mathrm{x}^2\)
• Group the constant terms: \(6 + (-2) = 4\)
• Final result: \(\mathrm{P} = 20\mathrm{x}^2 + 4\)

Answer: B) \(20\mathrm{x}^2 + 4\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students forget that perimeter means going around the entire rectangle, so they only add length + width once instead of using \(\mathrm{P} = 2\mathrm{l} + 2\mathrm{w}\).

They calculate: \(\mathrm{P} = (6\mathrm{x}^2 + 3) + (4\mathrm{x}^2 - 1) = 10\mathrm{x}^2 + 2\)

This leads them to select Choice A (\(10\mathrm{x}^2 + 2\))

Second Most Common Error:

Poor SIMPLIFY execution: Students make errors when applying the distributive property or combining like terms, especially with the negative terms.

Common mistakes include:

• Getting signs wrong: \(2(-1) = -2\), not +2
• Forgetting to distribute to all terms
• Errors in combining like terms

These calculation errors can lead to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students truly understand what perimeter means (going around the entire shape) and can execute multi-step algebraic simplification accurately. The key insight is recognizing that both length and width must be doubled before adding.