\((3\mathrm{x} - 2)(4\mathrm{x} + 5)\) The given expression is equivalent to 12x^2 + bx - 10, where b is a...

1. INFER the approach needed

• The problem tells us \((3x - 2)(4x + 5)\) is equivalent to \(12x^2 + bx - 10\)
• To find \(b\), we need to expand the left side and compare the x-coefficients

2. SIMPLIFY by expanding using the distributive property

• Apply FOIL (First, Outer, Inner, Last):
  • First: \(3x \times 4x = 12x^2\)
  • Outer: \(3x \times 5 = 15x\)
  • Inner: \(-2 \times 4x = -8x\)
  • Last: \(-2 \times 5 = -10\)

3. SIMPLIFY by combining like terms

• Collect the expanded terms: \(12x^2 + 15x - 8x - 10\)
• Combine the x terms: \(15x - 8x = 7x\)
• Final expanded form: \(12x^2 + 7x - 10\)

4. Compare coefficients

• Our expansion: \(12x^2 + 7x - 10\)
• Given form: \(12x^2 + bx - 10\)
• Therefore: \(b = 7\)

Answer: C) 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with sign errors: Students often make mistakes when distributing the negative sign from \((-2)\). They might calculate \((-2) \times (4x)\) as \(+8x\) instead of \(-8x\), or make errors when combining \(15x - 8x\).

For example, if they get \(-8x\) instead of \(+8x\), they would combine \(15x + 8x = 23x\), leading them to select Choice E (23).

Second Most Common Error:

Incomplete SIMPLIFY process: Some students correctly expand to get \(12x^2 + 15x - 8x - 10\) but then make arithmetic errors when combining \(15x - 8x\). They might miscalculate this as \(15x - 8x = 8x\) (subtracting incorrectly) or even \(15x - 8x = 1x\).

This could lead them to select Choice B (1) or cause confusion leading to guessing.

The Bottom Line:

This problem tests careful execution of algebraic expansion more than conceptual understanding. The strategy is straightforward, but students must be methodical about signs and arithmetic to avoid careless errors that lead to wrong answer choices.