When the expression \((3\mathrm{x} + 4)(2\mathrm{x} + \mathrm{k})\) is expanded, it is equivalent to 6x^2 + bx + 20. What...

1. TRANSLATE the problem information

• Given: \((3x + 4)(2x + k)\) expands to something equivalent to \(6x^2 + bx + 20\)
• What this means: The coefficients of like terms must be identical

2. SIMPLIFY by expanding the left expression

• Use distributive property on \((3x + 4)(2x + k)\):
  • First terms: \(3x \times 2x = 6x^2\)
  • Outer terms: \(3x \times k = 3kx\)
  • Inner terms: \(4 \times 2x = 8x\)
  • Last terms: \(4 \times k = 4k\)
• Combined: \(6x^2 + 3kx + 8x + 4k\)
• Collect like terms: \(6x^2 + (3k + 8)x + 4k\)

3. INFER the coefficient matching strategy

• Since \(6x^2 + (3k + 8)x + 4k = 6x^2 + bx + 20\)
• The coefficients of corresponding terms must be equal:
  • \(x^2\) terms: \(6 = 6\) ✓
  • \(x\) terms: \(3k + 8 = b\)
  • constant terms: \(4k = 20\)

4. SIMPLIFY to find the unknowns

• From \(4k = 20\): \(k = 5\)
• Substitute into \(b = 3k + 8\):
  \(b = 3(5) + 8\)
  \(b = 15 + 8\)
  \(b = 23\)

Answer: C) 23

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't recognize that "equivalent" means all corresponding coefficients must match. They might expand correctly but fail to set up the coefficient equations, leading to confusion about how to find b and potentially guessing randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes when expanding, such as getting \(6x^2 + 3kx + 8x + 4k = 6x^2 + (3k + 8)x + 4k\) wrong, or incorrectly combining the x terms. For example, they might write \(3k + 8 = 3(k + 8) = 3k + 24\), leading them toward Choice D (32) when they solve.

The Bottom Line:

This problem tests whether students understand that polynomial equivalence means identical coefficient structure, not just that two expressions have the same value for specific x values. The key insight is recognizing that matching coefficients creates the pathway to solving for unknowns.