If p = 3x + 4 and v = x + 5, which of the following is equivalent to pv...

1. TRANSLATE the problem information

• Given expressions:
  • \(\mathrm{p = 3x + 4}\)
  • \(\mathrm{v = x + 5}\)
• We need to find: \(\mathrm{pv - 2p + v}\)

2. TRANSLATE by substituting the given expressions

• Replace p and v in the target expression:
  \(\mathrm{pv - 2p + v}\) becomes \(\mathrm{(3x + 4)(x + 5) - 2(3x + 4) + (x + 5)}\)

3. SIMPLIFY by expanding the binomial product

• Use FOIL to expand \(\mathrm{(3x + 4)(x + 5)}\):
  • First: \(\mathrm{3x \times x = 3x^2}\)
  • Outer: \(\mathrm{3x \times 5 = 15x}\)
  • Inner: \(\mathrm{4 \times x = 4x}\)
  • Last: \(\mathrm{4 \times 5 = 20}\)
• Result: \(\mathrm{3x^2 + 15x + 4x + 20 = 3x^2 + 19x + 20}\)

4. SIMPLIFY by applying the distributive property

• Distribute \(\mathrm{-2}\) to \(\mathrm{(3x + 4)}\):
  \(\mathrm{-2(3x + 4) = -6x - 8}\)

5. SIMPLIFY by combining all terms

• Our expression becomes:
  \(\mathrm{3x^2 + 19x + 20 - 6x - 8 + x + 5}\)
• Combine like terms:
  • \(\mathrm{x^2}\) terms: \(\mathrm{3x^2}\)
  • x terms: \(\mathrm{19x - 6x + x = 14x}\)
  • Constants: \(\mathrm{20 - 8 + 5 = 17}\)

Answer: B. \(\mathrm{3x^2 + 14x + 17}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up the substitution but make sign errors when distributing the negative coefficient \(\mathrm{-2}\). They might distribute \(\mathrm{+2}\) instead of \(\mathrm{-2}\), getting \(\mathrm{+6x + 8}\) instead of \(\mathrm{-6x - 8}\). When they combine like terms, this gives them \(\mathrm{3x^2 + 26x + 33}\) instead of the correct answer.
This may lead them to select Choice D (\(\mathrm{3x^2 + 26x + 33}\)).

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly expand \(\mathrm{(3x + 4)(x + 5)}\) to get \(\mathrm{3x^2 + 19x + 20}\), but then stop there without completing the remaining operations (\(\mathrm{-2p + v}\)). They see this result matches one of the answer choices and select it.
This may lead them to select Choice C (\(\mathrm{3x^2 + 19x + 20}\)).

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to signs when distributing negative coefficients and methodical combining of like terms through multiple steps.