The expression \((3\mathrm{x} - 1)(2\mathrm{x}^2 + \mathrm{bx} + 4)\) is equivalent to 6x^3 + cx^2 + 10x - 4, where...

1. TRANSLATE the problem information

• Given information:
  • \((3\mathrm{x} - 1)(2\mathrm{x}^2 + \mathrm{bx} + 4)\) is equivalent to \(6\mathrm{x}^3 + \mathrm{cx}^2 + 10\mathrm{x} - 4\)
  • Need to find the value of c
• What this tells us: We need to expand the left side and match coefficients with the right side

2. SIMPLIFY by expanding the left side

• First, distribute 3x to each term in the second factor:
  \(3\mathrm{x}(2\mathrm{x}^2 + \mathrm{bx} + 4) = 6\mathrm{x}^3 + 3\mathrm{bx}^2 + 12\mathrm{x}\)
• Then, distribute -1 to each term in the second factor:
  \(-1(2\mathrm{x}^2 + \mathrm{bx} + 4) = -2\mathrm{x}^2 - \mathrm{bx} - 4\)
• Combining both parts:
  \(6\mathrm{x}^3 + 3\mathrm{bx}^2 + 12\mathrm{x} - 2\mathrm{x}^2 - \mathrm{bx} - 4\)

3. SIMPLIFY by collecting like terms

• Group terms with the same power:
  \(6\mathrm{x}^3 + (3\mathrm{b} - 2)\mathrm{x}^2 + (12 - \mathrm{b})\mathrm{x} - 4\)

4. INFER the coefficient matching strategy

• Since the expressions are equivalent, coefficients of like terms must be equal
• We have: \(6\mathrm{x}^3 + (3\mathrm{b} - 2)\mathrm{x}^2 + (12 - \mathrm{b})\mathrm{x} - 4 = 6\mathrm{x}^3 + \mathrm{cx}^2 + 10\mathrm{x} - 4\)
• This gives us the system:
  • x coefficient: \(12 - \mathrm{b} = 10\)
  • x² coefficient: \(3\mathrm{b} - 2 = \mathrm{c}\)

5. SIMPLIFY to solve for the unknowns

• From \(12 - \mathrm{b} = 10\):
  \(\mathrm{b} = 12 - 10 = 2\)
• Substitute \(\mathrm{b} = 2\) into the x² coefficient equation:
  \(\mathrm{c} = 3\mathrm{b} - 2\)
  \(\mathrm{c} = 3(2) - 2\)
  \(\mathrm{c} = 6 - 2\)
  \(\mathrm{c} = 4\)

Answer: C) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding or collecting like terms, leading to incorrect coefficients.

Common mistakes include:

• Forgetting the negative sign: writing \(+2\mathrm{x}^2\) instead of \(-2\mathrm{x}^2\)
• Sign errors when distributing: getting \(+\mathrm{bx}\) instead of \(-\mathrm{bx}\)
• Arithmetic mistakes when collecting: writing \(3\mathrm{b} + 2\) instead of \(3\mathrm{b} - 2\)

This may lead them to select Choice A (-2) or Choice B (2) depending on which coefficient they calculate incorrectly.

Second Most Common Error:

Poor INFER reasoning: Students try to solve for both b and c simultaneously without recognizing that one must be found first.

They might attempt to substitute expressions rather than solving the simpler equation \(12 - \mathrm{b} = 10\) first. This leads to unnecessary complexity and potential algebraic errors, causing them to get confused and abandon systematic solution.

The Bottom Line:

This problem requires careful algebraic manipulation and strategic thinking about which variable to solve for first. Success depends on methodical expansion and systematic coefficient matching rather than trying to take shortcuts.