2x^2 + 5x - 12 If the given expression is rewritten in the form \((2\mathrm{x} - 3)(\mathrm{x} + \mathrm{k})\), where...

1. TRANSLATE the problem requirement

• Given information:
  • Original expression: \(2\mathrm{x}^2 + 5\mathrm{x} - 12\)
  • Target form: \((2\mathrm{x} - 3)(\mathrm{x} + \mathrm{k})\)
  • Need to find: value of k
• What this tells us: The two expressions must be equivalent, so I can set them equal

2. INFER the solving approach

• Since the expressions are equivalent: \((2\mathrm{x} - 3)(\mathrm{x} + \mathrm{k}) = 2\mathrm{x}^2 + 5\mathrm{x} - 12\)
• Strategy: Expand the left side, then match coefficients of like terms

3. SIMPLIFY by expanding the left side

• Using FOIL on \((2\mathrm{x} - 3)(\mathrm{x} + \mathrm{k})\):
  • First: \(2\mathrm{x} \cdot \mathrm{x} = 2\mathrm{x}^2\)
  • Outer: \(2\mathrm{x} \cdot \mathrm{k} = 2\mathrm{kx}\)
  • Inner: \(-3 \cdot \mathrm{x} = -3\mathrm{x}\)
  • Last: \(-3 \cdot \mathrm{k} = -3\mathrm{k}\)
• Combined: \(2\mathrm{x}^2 + 2\mathrm{kx} - 3\mathrm{x} - 3\mathrm{k} = 2\mathrm{x}^2 + (2\mathrm{k} - 3)\mathrm{x} - 3\mathrm{k}\)

4. INFER coefficient matching strategy

• For \(2\mathrm{x}^2 + (2\mathrm{k} - 3)\mathrm{x} - 3\mathrm{k} = 2\mathrm{x}^2 + 5\mathrm{x} - 12\) to be true:
  • x² coefficients: \(2 = 2\) ✓ (already matches)
  • x coefficients: \(2\mathrm{k} - 3 = 5\)
  • constant terms: \(-3\mathrm{k} = -12\)

5. SIMPLIFY to solve for k

• Using the constant term equation: \(-3\mathrm{k} = -12\)
• Divide both sides by -3: \(\mathrm{k} = 4\)
• Verification using x coefficient: \(2\mathrm{k} - 3 = 5\)
  \(2(4) - 3 = 8 - 3 = 5\) ✓

Answer: k = 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors during expansion, particularly with the term \(-3 \cdot \mathrm{k} = -3\mathrm{k}\)

Students often write \((2\mathrm{x} - 3)(\mathrm{x} + \mathrm{k}) = 2\mathrm{x}^2 + 2\mathrm{kx} - 3\mathrm{x} + 3\mathrm{k}\) instead of \(2\mathrm{x}^2 + 2\mathrm{kx} - 3\mathrm{x} - 3\mathrm{k}\), making the constant term \(+3\mathrm{k}\) instead of \(-3\mathrm{k}\). This leads to the equation \(3\mathrm{k} = -12\), giving \(\mathrm{k} = -4\).

This may lead them to select an incorrect answer choice if -4 were available, or causes confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Arithmetic errors when solving linear equations

Students correctly set up \(-3\mathrm{k} = -12\) but make sign errors when dividing, arriving at \(\mathrm{k} = -4\) instead of \(\mathrm{k} = 4\), or they use the wrong coefficient equation and solve incorrectly.

This leads to confusion about the final answer and potential guessing.

The Bottom Line:

This problem tests careful algebraic manipulation more than conceptual understanding. Success depends on methodical expansion and accurate arithmetic throughout the solution process.