\((3\mathrm{x} - 2)^2 - (2\mathrm{x} + 1)(\mathrm{x} - 3)\)Which of the following is equivalent to the expression above?

1. INFER the solution strategy

• Given: \((3x - 2)^2 - (2x + 1)(x - 3)\)
• Strategy: Expand each part separately, then combine like terms
• This approach breaks down the complex expression into manageable pieces

2. SIMPLIFY the first part: \((3x - 2)^2\)

• Use the binomial square formula: \((a - b)^2 = a^2 - 2ab + b^2\)
• Where \(a = 3x\) and \(b = 2\):
  • \((3x)^2 = 9x^2\)
  • \(-2(3x)(2) = -12x\)
  • \((2)^2 = 4\)
• Result: \(9x^2 - 12x + 4\)

3. SIMPLIFY the second part: \((2x + 1)(x - 3)\)

• Use FOIL method:
  • First: \((2x)(x) = 2x^2\)
  • Outer: \((2x)(-3) = -6x\)
  • Inner: \((1)(x) = x\)
  • Last: \((1)(-3) = -3\)
• Combine: \(2x^2 - 6x + x - 3 = 2x^2 - 5x - 3\)

4. SIMPLIFY by substituting back and distributing

• Expression becomes: \((9x^2 - 12x + 4) - (2x^2 - 5x - 3)\)
• Distribute the negative sign carefully:
  \(9x^2 - 12x + 4 - 2x^2 + 5x + 3\)

5. SIMPLIFY by combining like terms

• \(x^2\) terms: \(9x^2 - 2x^2 = 7x^2\)
• \(x\) terms: \(-12x + 5x = -7x\)
• Constants: \(4 + 3 = 7\)
• Final result: \(7x^2 - 7x + 7\)

Answer: C. \(7x^2 - 7x + 7\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when distributing the negative sign

Students correctly expand both parts but make mistakes when handling the subtraction. They might write:
\(9x^2 - 12x + 4 - 2x^2 - 5x - 3\) (forgetting to distribute the negative to all terms)

This leads to: \(7x^2 - 17x + 1\), causing them to select Choice A (\(7x^2 - 17x + 1\))

Second Most Common Error:

Poor SIMPLIFY execution: Errors in the binomial square expansion

Students might incorrectly expand \((3x - 2)^2\) as \(9x^2 + 4\) (missing the middle term) or miscalculate the middle term as \(-6x\) instead of \(-12x\).

This leads to incorrect coefficients throughout and typically causes confusion, leading to guessing among the remaining choices.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to signs and methodical combination of like terms. The key is working step-by-step rather than trying to do multiple operations simultaneously.