A rectangular sheet of metal has length L = x + 6 and width W = 3x - 2. A...

1. TRANSLATE the problem information

• Given information:
  • Original sheet: length \(\mathrm{L = x + 6}\), width \(\mathrm{W = 3x - 2}\)
  • Strip removed: width 2, length \(\mathrm{L = x + 6}\)
  • Piece added: area = \(\mathrm{W = 3x - 2}\)
• What this tells us: We need to calculate initial area, subtract removed area, then add the welded piece area

2. INFER the solution approach

• The final area follows the pattern: \(\mathrm{Final = Initial - Removed + Added}\)
• We'll need to expand polynomials and combine like terms
• Work systematically through each component

3. Calculate the initial area

• Initial area = \(\mathrm{L \times W = (x + 6)(3x - 2)}\)
• SIMPLIFY using FOIL:
  • First: \(\mathrm{x \times 3x = 3x^2}\)
  • Outer: \(\mathrm{x \times (-2) = -2x}\)
  • Inner: \(\mathrm{6 \times 3x = 18x}\)
  • Last: \(\mathrm{6 \times (-2) = -12}\)
• Initial area = \(\mathrm{3x^2 - 2x + 18x - 12 = 3x^2 + 16x - 12}\)

4. Find the area removed

• Strip dimensions: width 2, length \(\mathrm{(x + 6)}\)
• Area removed = \(\mathrm{2(x + 6) = 2x + 12}\)

5. Identify area added

• Area added = \(\mathrm{W = 3x - 2}\)

6. SIMPLIFY to find final area

• Final area = \(\mathrm{(3x^2 + 16x - 12) - (2x + 12) + (3x - 2)}\)
• Distribute the negative sign: \(\mathrm{3x^2 + 16x - 12 - 2x - 12 + 3x - 2}\)
• SIMPLIFY by combining like terms:
  • \(\mathrm{x^2}\) terms: \(\mathrm{3x^2}\)
  • x terms: \(\mathrm{16x - 2x + 3x = 17x}\)
  • Constants: \(\mathrm{-12 - 12 - 2 = -26}\)
• Final area = \(\mathrm{3x^2 + 17x - 26}\)

Answer: C. \(\mathrm{3x^2 + 17x - 26}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "a rectangular piece with area W" and think they need to find dimensions rather than just using \(\mathrm{W = 3x - 2}\) as the area directly.

This leads them to get confused about what to add back, potentially trying to multiply W by something else or treating it as a dimension instead of an area. They might calculate incorrectly and select Choice A (\(\mathrm{3x^2 + 11x - 22}\)) or abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when distributing the negative through \(\mathrm{(2x + 12)}\), writing \(\mathrm{+2x + 12}\) instead of \(\mathrm{-2x - 12}\) in the final expression.

This error in combining like terms leads them to get \(\mathrm{3x^2 + 21x - 2}\), causing them to select Choice D (\(\mathrm{3x^2 + 21x - 2}\)).

The Bottom Line:

This problem requires careful translation of multiple area operations and meticulous algebraic simplification. Students must resist the urge to overcomplicate the "area W" component and focus on systematic sign management.