\((\mathrm{x} + 2)(\mathrm{x} + 3) = (\mathrm{x} - 2)(\mathrm{x} - 3) + 10\) Which of the following is a solution...

1. TRANSLATE the problem information

• Given equation: \((\mathrm{x} + 2)(\mathrm{x} + 3) = (\mathrm{x} - 2)(\mathrm{x} - 3) + 10\)
• Need to find which value satisfies this equation

2. INFER the solution strategy

• Both sides contain binomial products that need to be expanded
• After expansion, this will likely become a linear equation since the x² terms should cancel out
• Strategy: Expand both sides, then solve algebraically

3. SIMPLIFY by expanding the left side

• Use FOIL: \((\mathrm{x} + 2)(\mathrm{x} + 3)\)
  • First: \(\mathrm{x} \times \mathrm{x} = \mathrm{x}^2\)
  • Outer: \(\mathrm{x} \times 3 = 3\mathrm{x}\)
  • Inner: \(2 \times \mathrm{x} = 2\mathrm{x}\)
  • Last: \(2 \times 3 = 6\)
• Result: \(\mathrm{x}^2 + 3\mathrm{x} + 2\mathrm{x} + 6 = \mathrm{x}^2 + 5\mathrm{x} + 6\)

4. SIMPLIFY by expanding the right side

• First expand \((\mathrm{x} - 2)(\mathrm{x} - 3)\):
  • First: \(\mathrm{x} \times \mathrm{x} = \mathrm{x}^2\)
  • Outer: \(\mathrm{x} \times (-3) = -3\mathrm{x}\)
  • Inner: \((-2) \times \mathrm{x} = -2\mathrm{x}\)
  • Last: \((-2) \times (-3) = 6\)
• Result: \(\mathrm{x}^2 - 3\mathrm{x} - 2\mathrm{x} + 6 = \mathrm{x}^2 - 5\mathrm{x} + 6\)
• Add the constant: \(\mathrm{x}^2 - 5\mathrm{x} + 6 + 10 = \mathrm{x}^2 - 5\mathrm{x} + 16\)

5. SIMPLIFY the resulting equation

• We now have: \(\mathrm{x}^2 + 5\mathrm{x} + 6 = \mathrm{x}^2 - 5\mathrm{x} + 16\)
• Subtract \(\mathrm{x}^2\) from both sides: \(5\mathrm{x} + 6 = -5\mathrm{x} + 16\)
• Add \(5\mathrm{x}\) to both sides: \(10\mathrm{x} + 6 = 16\)
• Subtract 6 from both sides: \(10\mathrm{x} = 10\)
• Divide by 10: \(\mathrm{x} = 1\)

Answer: A. 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when expanding \((\mathrm{x} - 2)(\mathrm{x} - 3)\), particularly with the negative terms. Students often get confused about whether \((-2) \times (-3) = +6\) or \(-6\), or they might incorrectly handle the middle terms as \(+3\mathrm{x} + 2\mathrm{x}\) instead of \(-3\mathrm{x} - 2\mathrm{x}\).

This algebraic error propagates through the rest of the solution, leading to an incorrect final answer. This may lead them to select Choice B (0) or get confused and guess.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly expand the binomials but make errors when collecting like terms or solving the linear equation. For example, they might incorrectly combine \(5\mathrm{x}\) and \(-5\mathrm{x}\), or make arithmetic mistakes when isolating \(\mathrm{x}\).

This leads to confusion and potentially selecting any of the incorrect answer choices.

The Bottom Line:

This problem tests careful algebraic manipulation more than advanced concepts. The key challenge is maintaining accuracy through multiple steps of binomial expansion and equation solving, especially when dealing with negative terms.