Let \(\mathrm{P(x) = 3x^2 - 4}\) and \(\mathrm{Q(x) = x^2 - 5x + 1}\). Which of the following expressions is...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P(x) = 3x^2 - 4}\)
  • \(\mathrm{Q(x) = x^2 - 5x + 1}\)
  • Need to find \(\mathrm{P(x) - 2Q(x)}\)

• This means: subtract 2 times Q(x) from P(x)

2. SIMPLIFY by computing 2Q(x) first

• Multiply every term in Q(x) by 2:
  \(\mathrm{2Q(x) = 2(x^2 - 5x + 1) = 2x^2 - 10x + 2}\)

3. SIMPLIFY by setting up the subtraction

• Now we need: \(\mathrm{P(x) - 2Q(x)}\)
• Substitute: \(\mathrm{(3x^2 - 4) - (2x^2 - 10x + 2)}\)

4. SIMPLIFY by distributing the subtraction

• Distribute the negative sign to each term in the second polynomial:
  \(\mathrm{= 3x^2 - 4 - 2x^2 + 10x - 2}\)

5. SIMPLIFY by combining like terms

• Group similar terms:
  \(\mathrm{= (3x^2 - 2x^2) + 10x + (-4 - 2)}\)
  \(\mathrm{= x^2 + 10x - 6}\)

Answer: A (\(\mathrm{x^2 + 10x - 6}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when distributing the subtraction sign.

They correctly compute \(\mathrm{2Q(x) = 2x^2 - 10x + 2}\), but when subtracting, they write:
\(\mathrm{(3x^2 - 4) - (2x^2 - 10x + 2) = 3x^2 - 4 - 2x^2 - 10x - 2}\)

This gives them \(\mathrm{x^2 - 10x - 6}\) instead of \(\mathrm{x^2 + 10x - 6}\).
This may lead them to select Choice B (\(\mathrm{x^2 - 10x - 6}\)).

Second Most Common Error:

Weak SIMPLIFY skill: Students make errors when combining the x² terms.

They might incorrectly add the x² coefficients instead of subtracting: \(\mathrm{3x^2 + 2x^2 = 5x^2}\), leading to \(\mathrm{5x^2 + 10x - 6}\).
This may lead them to select Choice C (\(\mathrm{5x^2 - 10x - 2}\)).

The Bottom Line:

This problem tests careful execution of multiple algebraic steps. The key challenge is maintaining accuracy with signs and coefficients through several operations, especially when distributing subtraction across multiple terms.