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

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P(x) = x^2 - 3x + 5}\)
  • \(\mathrm{Q(x) = -x^2 + x - 4}\)
  • Need to find \(\mathrm{P(x) - 2Q(x)}\)

• This means we need to subtract twice \(\mathrm{Q(x)}\) from \(\mathrm{P(x)}\)

2. SIMPLIFY by computing 2Q(x) first

• Multiply \(\mathrm{Q(x)}\) by 2:
  \(\mathrm{2Q(x) = 2(-x^2 + x - 4)}\)

• Distribute the 2 to each term:
  \(\mathrm{2Q(x) = -2x^2 + 2x - 8}\)

3. SIMPLIFY the subtraction P(x) - 2Q(x)

• Set up the subtraction:
  \(\mathrm{P(x) - 2Q(x) = (x^2 - 3x + 5) - (-2x^2 + 2x - 8)}\)

• Distribute the negative sign to each term in \(\mathrm{2Q(x)}\):
  \(\mathrm{= x^2 - 3x + 5 + 2x^2 - 2x + 8}\)

4. SIMPLIFY by combining like terms

• Group terms by degree:
  \(\mathrm{= (x^2 + 2x^2) + (-3x - 2x) + (5 + 8)}\)

• Combine coefficients:
  \(\mathrm{= 3x^2 - 5x + 13}\)

Answer: D) \(\mathrm{3x^2 - 5x + 13}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students forget to distribute the negative sign when subtracting \(\mathrm{2Q(x)}\), treating subtraction as if parentheses weren't there.

Instead of \(\mathrm{P(x) - 2Q(x) = (x^2 - 3x + 5) - (-2x^2 + 2x - 8)}\), they compute:
\(\mathrm{P(x) - 2Q(x) = (x^2 - 3x + 5) - 2x^2 + 2x - 8}\)

This gives them: \(\mathrm{-x^2 - x - 3}\)
This may lead them to select Choice B (\(\mathrm{-x^2 - x - 3}\))

Second Most Common Error:

Weak SIMPLIFY execution: Students make sign errors when computing \(\mathrm{2Q(x)}\), particularly with the negative leading coefficient.

They might compute \(\mathrm{2Q(x) = 2x^2 + 2x - 8}\) instead of \(\mathrm{-2x^2 + 2x - 8}\), leading to various incorrect combinations when they subtract.
This causes confusion in the combining step and leads to guessing among the remaining choices.

The Bottom Line:

This problem tests careful algebraic manipulation with multiple opportunities for sign errors. Success requires systematic attention to distributing negative signs and combining like terms methodically.