Question:A square has side length 2x - 1 units.A rectangle has length x - 3 units and width x +...

1. TRANSLATE the problem information

• Given information:
  • Square has side length: \((2x - 1)\) units
  • Rectangle has length: \((x - 3)\) units and width: \((x + 2)\) units
  • Need: Sum of both areas as a polynomial

2. INFER the approach

• We need to find each area separately using the appropriate formulas, then add them
• Square area uses \(A = s²\), rectangle area uses \(A = \text{length} \times \text{width}\)
• Each will give us a polynomial that we then combine

3. SIMPLIFY to find the square's area

• Square area = (side length)² = \((2x - 1)²\)
• Expand using \((a - b)² = a² - 2ab + b²\):

\((2x - 1)² = (2x)² - 2(2x)(1) + (1)²\)
\(= 4x² - 4x + 1\)

4. SIMPLIFY to find the rectangle's area

• Rectangle area = length × width = \((x - 3)(x + 2)\)
• Use FOIL method:
  • First: \(x \cdot x = x²\)
  • Outer: \(x \cdot 2 = 2x\)
  • Inner: \((-3) \cdot x = -3x\)
  • Last: \((-3) \cdot 2 = -6\)
• Combine: \(x² + 2x - 3x - 6\)
  \(= x² - x - 6\)

5. SIMPLIFY to find the total area

• Sum = Square area + Rectangle area
• Sum = \((4x² - 4x + 1) + (x² - x - 6)\)
• Combine like terms:
  • \(x²\) terms: \(4x² + x² = 5x²\)
  • \(x\) terms: \(-4x + (-x) = -5x\)
  • Constants: \(1 + (-6) = -5\)
• Final result: \(5x² - 5x - 5\)

Answer: B. \(5x² - 5x - 5\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Incorrectly expanding \((2x - 1)²\)

Many students forget the middle term when expanding a binomial square, writing \((2x - 1)² = 4x² + 1\) instead of \(4x² - 4x + 1\). They either completely omit the \(-4x\) term or get the sign wrong.

This leads to a total of \((4x² + 1) + (x² - x - 6) = 5x² - x - 5\), causing them to select Choice A (\(5x² - x - 7\)) after further arithmetic errors, or Choice C (\(5x² - 3x - 5\)) if they make additional sign errors.

Second Most Common Error:

Poor SIMPLIFY execution: Errors in FOIL multiplication

Students may correctly expand the square but make mistakes when applying FOIL to \((x - 3)(x + 2)\). Common errors include getting confused about the signs of the middle terms or forgetting to combine \(-3x\) and \(+2x\) properly.

This typically leads to wrong coefficients in the final polynomial, causing them to select Choice D (\(5x² - 5x + 7\)) if they get the constant term wrong.

The Bottom Line:

This problem tests careful algebraic manipulation across multiple steps. Success requires methodical expansion of both expressions and systematic combination of like terms without sign errors.