The expression \((4\mathrm{x} - 5)(7\mathrm{x} + 3)\) is equivalent to ax^2 + bx + c, where a, b, and c...

1. TRANSLATE the problem information

• Given information:
  • Expression \((4\mathrm{x} - 5)(7\mathrm{x} + 3)\) equals \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\)
  • Need to find \(\mathrm{a} + \mathrm{c}\) (coefficient of x² plus constant term)
• This tells us we need to expand the product and identify specific coefficients

2. SIMPLIFY by expanding the binomial product

• Use FOIL method to multiply \((4\mathrm{x} - 5)(7\mathrm{x} + 3)\):
  • First terms: \((4\mathrm{x})(7\mathrm{x}) = 28\mathrm{x}^2\)
  • Outer terms: \((4\mathrm{x})(3) = 12\mathrm{x}\)
  • Inner terms: \((-5)(7\mathrm{x}) = -35\mathrm{x}\)
  • Last terms: \((-5)(3) = -15\)

3. SIMPLIFY by combining like terms

• Combine all terms: \(28\mathrm{x}^2 + 12\mathrm{x} - 35\mathrm{x} - 15\)
• Combine the x terms: \(12\mathrm{x} - 35\mathrm{x} = -23\mathrm{x}\)
• Final expanded form: \(28\mathrm{x}^2 - 23\mathrm{x} - 15\)

4. TRANSLATE coefficients from expanded form

• From \(28\mathrm{x}^2 - 23\mathrm{x} - 15\) compared to \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\):
  • \(\mathrm{a} = 28\) (coefficient of x²)
  • \(\mathrm{b} = -23\) (coefficient of x)
  • \(\mathrm{c} = -15\) (constant term)

5. Calculate the final answer

• \(\mathrm{a} + \mathrm{c} = 28 + (-15) = 28 - 15 = 13\)

Answer: B) 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when combining like terms, particularly with \(12\mathrm{x} - 35\mathrm{x} = -23\mathrm{x}\), or make arithmetic mistakes during FOIL expansion.

Common mistakes include:

• Writing \(12\mathrm{x} - 35\mathrm{x}\) as \(23\mathrm{x}\) instead of \(-23\mathrm{x}\)
• Sign errors like \((-5)(3) = +15\) instead of \(-15\)
• Forgetting negative signs from the original expression

This may lead them to calculate incorrect values for a and c, potentially selecting Choice A (5) or Choice C (28) if they get confused about which coefficients to add.

The Bottom Line:

This problem tests careful algebraic manipulation - success depends on systematic FOIL expansion and meticulous attention to signs when combining terms.