The expression \((3\mathrm{x} - 23)(19\mathrm{x} + 6)\) is equivalent to the expression ax^2 + bx + c, where a, b,...

1. TRANSLATE the problem information

• Given: \((3x - 23)(19x + 6)\) is equivalent to \(ax^2 + bx + c\)
• Find: The value of coefficient b

2. SIMPLIFY by expanding the product

• Apply distributive property to \((3x - 23)(19x + 6)\):
  • First terms: \((3x)(19x) = 57x^2\)
  • Outer terms: \((3x)(6) = 18x\)
  • Inner terms: \((-23)(19x) = -437x\)
  • Last terms: \((-23)(6) = -138\)
• This gives us: \(57x^2 + 18x - 437x - 138\)

3. SIMPLIFY by combining like terms

• Combine the x terms: \(18x + (-437x) = 18x - 437x = -419x\)
• Final expanded form: \(57x^2 - 419x - 138\)

4. TRANSLATE to identify coefficient b

• Comparing \(57x^2 - 419x - 138\) to \(ax^2 + bx + c\):
  • The coefficient of \(x^2\) is \(a = 57\)
  • The coefficient of x is \(b = -419\)
  • The constant term is \(c = -138\)

Answer: -419

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when applying the distributive property, particularly with the negative terms \((-23)(19x)\) and \((-23)(6)\). They might calculate \((-23)(19x)\) as \(+437x\) instead of \(-437x\), or make errors when combining \(18x - 437x\).

This leads to incorrect coefficient values and selecting a wrong answer.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly expand but make arithmetic errors when combining like terms, such as calculating \(18 - 437\) incorrectly or forgetting to include the negative sign in the final coefficient.

This causes confusion about the correct value of b and may lead to guessing among similar answer choices.

The Bottom Line:

This problem tests careful algebraic manipulation with attention to signs. Success requires systematic application of distributive property followed by meticulous combination of like terms, especially when negative coefficients are involved.