Which expression is equivalent to \(5\mathrm{x}(2\mathrm{x} + 3) - 3\mathrm{x}(\mathrm{x} + 4)\)?

1. INFER the solution strategy

• Given: \(\mathrm{5x(2x + 3) - 3x(x + 4)}\)
• Strategy: We need to distribute each term first, then combine like terms
• This follows the order of operations - handle parentheses through distribution first

2. SIMPLIFY by distributing the first term

• Distribute 5x to each term in (2x + 3):
  • \(\mathrm{5x \cdot 2x = 10x^2}\)
  • \(\mathrm{5x \cdot 3 = 15x}\)
• Result: \(\mathrm{5x(2x + 3) = 10x^2 + 15x}\)

3. SIMPLIFY by distributing the second term

• Distribute -3x to each term in (x + 4):
  • \(\mathrm{(-3x) \cdot x = -3x^2}\)
  • \(\mathrm{(-3x) \cdot 4 = -12x}\)
• Result: \(\mathrm{-3x(x + 4) = -3x^2 - 12x}\)

4. SIMPLIFY by combining the distributed expressions

• Full expression: \(\mathrm{10x^2 + 15x - 3x^2 - 12x}\)
• Group like terms: \(\mathrm{(10x^2 - 3x^2) + (15x - 12x)}\)

5. SIMPLIFY by combining like terms

• x² terms: \(\mathrm{10x^2 - 3x^2 = 7x^2}\)
• x terms: \(\mathrm{15x - 12x = 3x}\)
• Final result: \(\mathrm{7x^2 + 3x}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with sign errors: Students correctly understand they need to distribute but make sign errors, particularly when handling the negative sign in front of 3x. They might treat \(\mathrm{-3x(x + 4)}\) as \(\mathrm{3x^2 + 12x}\) instead of \(\mathrm{-3x^2 - 12x}\).

When they calculate \(\mathrm{(10x^2 + 3x^2) + (15x + 12x)}\), they get \(\mathrm{13x^2 + 27x}\), leading them to select Choice C (\(\mathrm{13x^2 + 27x}\)).

Second Most Common Error:

Poor SIMPLIFY execution in combining like terms: Students distribute correctly but make arithmetic errors when combining like terms. They might calculate the x terms as \(\mathrm{15x + 12x = 27x}\) instead of \(\mathrm{15x - 12x = 3x}\), while correctly getting \(\mathrm{7x^2}\) for the x² coefficient.

This leads them to select Choice A (\(\mathrm{7x^2 + 27x}\)).

The Bottom Line:

This problem tests systematic algebraic manipulation skills. The key challenge is maintaining accuracy through multiple steps while carefully tracking positive and negative signs throughout the distribution and combination process.