Which of the following expressions is equivalent to \(2(\mathrm{ab} - 3) + 2\)?

1. INFER the approach needed

• We have the expression \(2(\mathrm{ab} - 3) + 2\)
• We need to simplify this by first applying the distributive property, then combining any like terms

2. SIMPLIFY using the distributive property

• Apply the distributive property to \(2(\mathrm{ab} - 3)\):
  • \(2(\mathrm{ab} - 3) = 2(\mathrm{ab}) + 2(-3) = 2\mathrm{ab} - 6\)
• Now our expression is: \(2\mathrm{ab} - 6 + 2\)

3. SIMPLIFY by combining like terms

• Combine the constant terms: \(-6 + 2 = -4\)
• Final result: \(2\mathrm{ab} - 4\)

Answer: B. \(2\mathrm{ab} - 4\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students don't fully apply the distributive property to both terms inside the parentheses.

They might distribute the 2 to ab but forget to distribute it to -3, thinking: \(2(\mathrm{ab} - 3) = 2\mathrm{ab} - 3\). Then adding the +2 at the end gives \(2\mathrm{ab} - 3 + 2 = 2\mathrm{ab} - 1\).

This may lead them to select Choice A (\(2\mathrm{ab} - 1\)).

The Bottom Line:

The distributive property must be applied completely to every term inside parentheses before moving to the next step. Missing even one part of the distribution creates a cascade of errors in the final answer.