Which of the following is equivalent to the expression \((3\mathrm{t} + 2\mathrm{s})^2 - (\mathrm{t} + 2\mathrm{s})(\mathrm{t} - 4\mathrm{s})\)?8t^2 +...

1. INFER the solution strategy

• This problem requires expanding two expressions separately, then subtracting
• We have: \((3\mathrm{t} + 2\mathrm{s})² − (\mathrm{t} + 2\mathrm{s})(\mathrm{t} − 4\mathrm{s})\)
• Strategy: Expand each part completely, then combine like terms

2. SIMPLIFY the first expression using perfect square formula

• \((3\mathrm{t} + 2\mathrm{s})² = (3\mathrm{t})² + 2(3\mathrm{t})(2\mathrm{s}) + (2\mathrm{s})²\)
• \(= 9\mathrm{t}² + 12\mathrm{ts} + 4\mathrm{s}²\)

3. SIMPLIFY the second expression using FOIL

• \((\mathrm{t} + 2\mathrm{s})(\mathrm{t} − 4\mathrm{s}) = \mathrm{t}² - 4\mathrm{ts} + 2\mathrm{st} - 8\mathrm{s}²\)
• Combine the middle terms: \(-4\mathrm{ts} + 2\mathrm{ts} = -2\mathrm{ts}\)
• \(= \mathrm{t}² - 2\mathrm{ts} - 8\mathrm{s}²\)

4. SIMPLIFY the subtraction carefully

• \((9\mathrm{t}² + 12\mathrm{ts} + 4\mathrm{s}²) - (\mathrm{t}² - 2\mathrm{ts} - 8\mathrm{s}²)\)
• Distribute the negative sign: \(9\mathrm{t}² + 12\mathrm{ts} + 4\mathrm{s}² - \mathrm{t}² + 2\mathrm{ts} + 8\mathrm{s}²\)
• SIMPLIFY by combining like terms:
  • t² terms: \(9\mathrm{t}² - \mathrm{t}² = 8\mathrm{t}²\)
  • ts terms: \(12\mathrm{ts} + 2\mathrm{ts} = 14\mathrm{ts}\)
  • s² terms: \(4\mathrm{s}² + 8\mathrm{s}² = 12\mathrm{s}²\)

Answer: (D) \(8\mathrm{t}² + 14\mathrm{ts} + 12\mathrm{s}²\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when distributing the negative during subtraction

Students correctly expand both expressions but make mistakes like:

\((9\mathrm{t}² + 12\mathrm{ts} + 4\mathrm{s}²) - (\mathrm{t}² - 2\mathrm{ts} - 8\mathrm{s}²) = 9\mathrm{t}² + 12\mathrm{ts} + 4\mathrm{s}² - \mathrm{t}² - 2\mathrm{ts} - 8\mathrm{s}²\)

This gives them \(8\mathrm{t}² + 10\mathrm{ts} - 4\mathrm{s}²\), leading them to select Choice (A).

Second Most Common Error:

Weak SIMPLIFY skill: Errors in expanding the perfect square

Students might expand \((3\mathrm{t} + 2\mathrm{s})²\) incorrectly as \(9\mathrm{t}² + 4\mathrm{s}²\) (forgetting the middle term) or get the middle term coefficient wrong. Combined with other small errors, this leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests careful algebraic manipulation across multiple steps. Success requires systematic expansion of each expression followed by meticulous attention to signs when combining terms.