Let \(\mathrm{P(x) = 4x(3x - 2) + 5x^2 - 7x}\). Which expression is equivalent to \(\mathrm{P(x)}\)?

1. TRANSLATE the problem information

• Given: \(\mathrm{P(x) = 4x(3x - 2) + 5x^2 - 7x}\)
• Find: Which expression is equivalent to \(\mathrm{P(x)}\)
• This means we need to simplify the given expression to match one of the answer choices

2. SIMPLIFY by applying the distributive property

• Focus on \(\mathrm{4x(3x - 2)}\) first:
  • \(\mathrm{4x \cdot 3x = 12x^2}\)
  • \(\mathrm{4x \cdot (-2) = -8x}\)
• So \(\mathrm{4x(3x - 2) = 12x^2 - 8x}\)

3. SIMPLIFY by rewriting the full expression

• \(\mathrm{P(x) = (12x^2 - 8x) + 5x^2 - 7x}\)
• \(\mathrm{P(x) = 12x^2 - 8x + 5x^2 - 7x}\)

4. SIMPLIFY by combining like terms

• Group \(\mathrm{x^2}\) terms: \(\mathrm{12x^2 + 5x^2 = 17x^2}\)
• Group \(\mathrm{x}\) terms: \(\mathrm{-8x + (-7x) = -8x - 7x = -15x}\)
• Final result: \(\mathrm{P(x) = 17x^2 - 15x}\)

Answer: D (\(\mathrm{17x^2 - 15x}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when distributing or combining terms.

For example, they might calculate \(\mathrm{4x(3x - 2)}\) as \(\mathrm{12x^2 + 8x}\) instead of \(\mathrm{12x^2 - 8x}\), forgetting that \(\mathrm{4x \cdot (-2) = -8x}\), not \(\mathrm{+8x}\). Or when combining \(\mathrm{-8x - 7x}\), they might get \(\mathrm{-1x}\) instead of \(\mathrm{-15x}\).

This may lead them to select Choice A (\(\mathrm{16x^2 - 15x}\)) or Choice B (\(\mathrm{17x^2 - 9x}\)).

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly distribute but make computational errors when combining like terms.

They might correctly get \(\mathrm{12x^2 - 8x + 5x^2 - 7x}\) but then add the \(\mathrm{x^2}\) coefficients incorrectly (\(\mathrm{12 + 5 = 16}\) instead of \(\mathrm{17}\)) or combine the \(\mathrm{x}\) terms incorrectly (\(\mathrm{-8 - 7 = -9}\) instead of \(\mathrm{-15}\)).

This leads them to select Choice A (\(\mathrm{16x^2 - 15x}\)) or Choice C (\(\mathrm{17x^2 - x}\)).

The Bottom Line:

This problem tests careful algebraic manipulation. Success requires systematic distribution and methodical combination of like terms while maintaining attention to positive and negative signs throughout the process.