Which expression is equivalent to \(4(\mathrm{a}^2 + 6)\)?

1. SIMPLIFY using the distributive property

• Given expression: \(4(\mathrm{a}^2 + 6)\)
• Apply the distributive property: multiply 4 by each term inside the parentheses
• \(4(\mathrm{a}^2 + 6) = 4 \cdot \mathrm{a}^2 + 4 \cdot 6\)

2. SIMPLIFY the multiplication

• Calculate each product:
  • \(4 \cdot \mathrm{a}^2 = 4\mathrm{a}^2\)
  • \(4 \cdot 6 = 24\)
• Combine the results: \(4\mathrm{a}^2 + 24\)

Answer: A. \(4\mathrm{a}^2 + 24\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students fail to apply the distributive property completely, only multiplying the 4 by the first term \(\mathrm{a}^2\) and leaving the 6 unchanged.

Their reasoning: "I need to multiply 4 times \(\mathrm{a}^2\), which gives me \(4\mathrm{a}^2\), and then I just add the 6 that's already there."

This leads them to select Choice C \(4\mathrm{a}^2 + 6\).

The Bottom Line:

This problem tests whether students truly understand that when you have a number multiplying a parenthetical expression, that number must be distributed to every term inside the parentheses, not just the first one.