A rectangle has length x + 3 units and width 5 units. Which of the following expressions represents the perimeter...

1. TRANSLATE the problem information

• Given information:
  • Rectangle length: \(\mathrm{x + 3}\) units
  • Rectangle width: \(5\) units
  • Need: Perimeter in factored form

2. INFER the approach

• Use the perimeter formula for rectangles: \(\mathrm{P = 2(length) + 2(width)}\)
• Substitute our known values, then simplify and factor

3. SIMPLIFY by substituting and distributing

• \(\mathrm{P = 2(x + 3) + 2(5)}\)
• \(\mathrm{P = 2x + 6 + 10}\)
• \(\mathrm{P = 2x + 16}\)

4. SIMPLIFY by factoring out the greatest common factor

• Both terms have a common factor of 2
• \(\mathrm{P = 2(x + 8)}\)

Answer: C. \(\mathrm{2(x + 8)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{P = 2(x + 3) + 2(5)}\) and distribute to get \(\mathrm{P = 2x + 6 + 10}\), but then make arithmetic errors when combining the constant terms. Instead of \(\mathrm{6 + 10 = 16}\), they might calculate \(\mathrm{6 + 10 = 14}\) or some other incorrect sum.

This leads to incorrect expressions that don't match any of the given factored forms, causing confusion and potentially leading them to select Choice A: \(\mathrm{2(x + 3)}\) as the "simplest looking" option.

Second Most Common Error:

Missing conceptual knowledge of perimeter formula: Students might confuse perimeter with area and attempt to use \(\mathrm{P = length \times width}\) instead of \(\mathrm{P = 2(length) + 2(width)}\). This fundamental misunderstanding derails the entire solution from the start.

This causes them to get stuck early and resort to guessing among the answer choices.

The Bottom Line:

This problem tests both computational accuracy and conceptual understanding. Students must correctly apply the perimeter formula AND execute multiple algebraic steps without arithmetic errors to reach the factored form.