The expression \((\mathrm{x} + 2)(3\mathrm{x} - 1) + 7\mathrm{x} + 15\) can be written in the form ax^2 + bx...

1. TRANSLATE the problem information

• Given: Expression \((x + 2)(3x - 1) + 7x + 15\) needs to be written as \(ax² + bx + c\)
• Find: The value of \(a + b + c\)

2. SIMPLIFY by expanding the binomial product first

• Use the distributive property on \((x + 2)(3x - 1)\):
  • \(x(3x - 1) = 3x² - x\)
  • \(2(3x - 1) = 6x - 2\)
• Combined: \((x + 2)(3x - 1) = 3x² - x + 6x - 2 = 3x² + 5x - 2\)

3. SIMPLIFY by adding the remaining terms

• Now we have: \(3x² + 5x - 2 + 7x + 15\)
• Combine like terms:
  • \(x²\) terms: \(3x²\) (only one)
  • \(x\) terms: \(5x + 7x = 12x\)
  • Constant terms: \(-2 + 15 = 13\)
• Final form: \(3x² + 12x + 13\)

4. TRANSLATE to identify coefficients

• In the form \(ax² + bx + c\): \(a = 3, b = 12, c = 13\)
• Therefore: \(a + b + c = 3 + 12 + 13 = 28\)

Answer: C) 28

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make errors when expanding \((x + 2)(3x - 1)\), particularly with the middle terms or signs.

For example, they might get \(3x² - x + 6x - 2 = 3x² + 5x - 2\) wrong by calculating the middle terms incorrectly as \(3x² + 7x - 2\) or \(3x² + 4x - 2\). This leads to different values for the x-coefficient, resulting in a wrong final sum. They might select Choice B (27) or Choice A (15) depending on their specific error.

Second Most Common Error:

Arithmetic mistakes in SIMPLIFY: Students correctly expand the binomial but make errors when combining the x-terms \((5x + 7x)\) or constant terms \((-2 + 15)\).

Getting \(5x + 7x = 11x\) instead of \(12x\) would give them \(a = 3, b = 11, c = 13\), leading to \(a + b + c = 27\). This may lead them to select Choice B (27).

The Bottom Line:

This problem requires careful step-by-step algebraic manipulation. Success depends on systematic expansion of the binomial product and methodical combining of like terms—both areas where small errors compound into wrong final answers.