\((\mathrm{x} - 11\mathrm{y})(2\mathrm{x} - 3\mathrm{y}) - 12\mathrm{y}(-2\mathrm{x} + 3\mathrm{y})\) Which of the following is equivalent to the exp...

1. INFER the approach needed

• We have two main parts to handle:
  • A binomial product: \(\mathrm{(x - 11y)(2x - 3y)}\)
  • A distribution: \(\mathrm{-12y(-2x + 3y)}\)
• Strategy: Expand each part separately, then combine like terms

2. SIMPLIFY the first binomial product

• Using FOIL on \(\mathrm{(x - 11y)(2x - 3y)}\):
  • First terms: \(\mathrm{x \times 2x = 2x^2}\)
  • Outer terms: \(\mathrm{x \times (-3y) = -3xy}\)
  • Inner terms: \(\mathrm{(-11y) \times 2x = -22xy}\)
  • Last terms: \(\mathrm{(-11y) \times (-3y) = +33y^2}\)
• Combined: \(\mathrm{2x^2 - 3xy - 22xy + 33y^2 = 2x^2 - 25xy + 33y^2}\)

3. SIMPLIFY the distribution

• For \(\mathrm{-12y(-2x + 3y)}\):
  • \(\mathrm{-12y \times (-2x) = +24xy}\)
  • \(\mathrm{-12y \times 3y = -36y^2}\)
• Result: \(\mathrm{24xy - 36y^2}\)

4. SIMPLIFY by combining all terms

• Full expression: \(\mathrm{2x^2 - 25xy + 33y^2 + 24xy - 36y^2}\)
• Group like terms:
  • \(\mathrm{x^2}\) terms: \(\mathrm{2x^2}\)
  • xy terms: \(\mathrm{-25xy + 24xy = -xy}\)
  • \(\mathrm{y^2}\) terms: \(\mathrm{33y^2 - 36y^2 = -3y^2}\)

Answer: B. \(\mathrm{2x^2 - xy - 3y^2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make sign errors when distributing the \(\mathrm{-12y}\) term or when combining like terms. For example, they might incorrectly calculate \(\mathrm{-12y \times (-2x) = -24xy}\) instead of \(\mathrm{+24xy}\), or make mistakes combining the xy terms \(\mathrm{(-25xy + 24xy)}\). These calculation errors can lead to selecting Choice C (\(\mathrm{2x^2 + 24xy + 36y^2}\)) or Choice D (\(\mathrm{2x^2 - 49xy + 69y^2}\)).

Second Most Common Error:

Poor INFER reasoning: Some students try to take shortcuts by looking for patterns in the answer choices rather than systematically expanding. This can lead them to incorrectly assume the expression simplifies to a much simpler form, causing them to select Choice A (\(\mathrm{x - 23y}\)).

The Bottom Line:

This problem requires careful attention to signs and systematic algebraic manipulation. Success depends on methodically expanding each part and accurately combining like terms rather than rushing through the calculations.