The expression \((3\mathrm{x} - 5)(4\mathrm{x} + 2)\) is a polynomial. If the expression is written in the form ax^2 +...

1. TRANSLATE the problem requirements

• Given: The expression \((3\mathrm{x} - 5)(4\mathrm{x} + 2)\)
• Need to find: The value of \(\mathrm{b}\) when written as \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\)
• What this means: We need to expand the product and identify the coefficient of the \(\mathrm{x}\) term

2. SIMPLIFY by expanding the binomial product

• Use the distributive property: \((3\mathrm{x} - 5)(4\mathrm{x} + 2) = 3\mathrm{x}(4\mathrm{x} + 2) - 5(4\mathrm{x} + 2)\)
• Distribute each term:
  • \(3\mathrm{x}(4\mathrm{x} + 2) = 12\mathrm{x}^2 + 6\mathrm{x}\)
  • \(-5(4\mathrm{x} + 2) = -20\mathrm{x} - 10\)
• Combine: \(12\mathrm{x}^2 + 6\mathrm{x} - 20\mathrm{x} - 10\)

3. SIMPLIFY by combining like terms

• Group similar terms: \(12\mathrm{x}^2 + (6\mathrm{x} - 20\mathrm{x}) - 10\)
• Combine the \(\mathrm{x}\) terms: \(6\mathrm{x} - 20\mathrm{x} = -14\mathrm{x}\)
• Final expanded form: \(12\mathrm{x}^2 - 14\mathrm{x} - 10\)

4. TRANSLATE to identify the coefficient

• In standard form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), we have:
  • \(\mathrm{a} = 12\) (coefficient of \(\mathrm{x}^2\))
  • \(\mathrm{b} = -14\) (coefficient of \(\mathrm{x}\))
  • \(\mathrm{c} = -10\) (constant term)

Answer: B) −14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the negative term \(-5(4\mathrm{x} + 2)\)

They might calculate: \(-5(4\mathrm{x} + 2) = -20\mathrm{x} + 10\) (incorrect positive 10)

This gives: \(12\mathrm{x}^2 + 6\mathrm{x} - 20\mathrm{x} + 10 = 12\mathrm{x}^2 - 14\mathrm{x} + 10\)

They still get \(\mathrm{b} = -14\) correctly, but this type of error pattern shows up in more complex problems and builds bad habits.

Second Most Common Error:

Inadequate SIMPLIFY execution: Arithmetic mistakes when combining like terms \(6\mathrm{x} - 20\mathrm{x}\)

Some students might compute: \(6\mathrm{x} - 20\mathrm{x} = -26\mathrm{x}\) instead of \(-14\mathrm{x}\)

This leads them to select Choice A (−20) if they confuse this with another coefficient, or causes confusion and guessing.

The Bottom Line:

This problem tests fundamental algebraic manipulation skills. Success requires careful attention to signs during distribution and accurate arithmetic when combining like terms. Students who rush through the expansion often make preventable errors that compound throughout the solution.