The expression \((2\mathrm{x} - 3)(\mathrm{x} + \mathrm{k}) + \mathrm{j}\) equals 2x^2 + px + q, where j, k, p, and...

1. TRANSLATE the problem information

• Given information:
  • \((2x - 3)(x + k) + j\) equals \(2x^2 + px + q\)
  • \(k = 5\) and \(j = -4\)
  • Need to find \(p + q\)
• What this tells us: We need to expand the left side, then match coefficients with the right side.

2. TRANSLATE by substituting known values

• Replace \(k\) with \(5\) and \(j\) with \(-4\):
  \((2x - 3)(x + 5) + (-4) = 2x^2 + px + q\)

3. SIMPLIFY by expanding the binomial

• Use FOIL to expand \((2x - 3)(x + 5)\):
  • First: \(2x \cdot x = 2x^2\)
  • Outer: \(2x \cdot 5 = 10x\)
  • Inner: \(-3 \cdot x = -3x\)
  • Last: \(-3 \cdot 5 = -15\)
• Combine:
  \(2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15\)

4. SIMPLIFY by adding the constant term

• Add \(j = -4\):
  \(2x^2 + 7x - 15 + (-4) = 2x^2 + 7x - 19\)

5. INFER by comparing coefficients

• We have: \(2x^2 + 7x - 19 = 2x^2 + px + q\)
• Comparing coefficients:
  • \(x\) coefficient: \(7 = p\), so \(p = 7\)
  • Constant term: \(-19 = q\), so \(q = -19\)

6. SIMPLIFY the final calculation

• \(p + q = 7 + (-19) = -12\)

Answer: B. -12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the FOIL expansion or when combining like terms.

For example, they might calculate:

• \((2x - 3)(x + 5) = 2x^2 + 10x - 3x - 15 = 2x^2 + 13x - 15\) (adding \(10x + 3x\) instead of \(10x - 3x\))

This leads to \(p = 13\) and \(q = -19\), giving \(p + q = -6\). Since \(-6\) isn't among the choices, this causes confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students forget to add \(j = -4\) to their expanded expression.

They correctly expand \((2x - 3)(x + 5) = 2x^2 + 7x - 15\), but then compare this directly to \(2x^2 + px + q\), getting \(p = 7\) and \(q = -15\). This gives \(p + q = -8\), which also isn't among the choices, leading to random selection.

The Bottom Line:

This problem tests whether students can systematically work through polynomial expansion while keeping track of all given values. The key insight is that every piece of given information (\(k = 5\) AND \(j = -4\)) must be incorporated into the final expression before comparing coefficients.