Which expression is equivalent to \(15 - (4 + \mathrm{y}^2)\)?

1. INFER the approach needed

• I see parentheses with a subtraction sign in front: \(15 - (4 + \mathrm{y}^2)\)
• Strategy: Distribute the negative sign to eliminate parentheses, then combine like terms

2. SIMPLIFY by distributing the negative sign

• Apply distributive property: \(15 - (4 + \mathrm{y}^2) = 15 - 4 - \mathrm{y}^2\)
• The negative sign applies to both terms inside the parentheses

3. SIMPLIFY by combining like terms

• Combine the constants: \(15 - 4 = 11\)
• Result: \(11 - \mathrm{y}^2\)

4. TRANSLATE to match answer format

• Rewrite with variable term first: \(-\mathrm{y}^2 + 11\)
• This matches answer choice A

Answer: A (\(-\mathrm{y}^2 + 11\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Not distributing the negative sign to the \(\mathrm{y}^2\) term

Students might think: "15 minus the quantity \((4 + \mathrm{y}^2)\) means I subtract 4 but not the \(\mathrm{y}^2\)"
This leads to: \(15 - 4 + \mathrm{y}^2 = 11 + \mathrm{y}^2\)

This may lead them to select Choice C (\(\mathrm{y}^2 + 11\))

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when combining constants

Students correctly distribute to get \(15 - 4 - \mathrm{y}^2\), but then calculate \(15 - 4 = 19\) instead of 11
This gives them: \(19 - \mathrm{y}^2 = -\mathrm{y}^2 + 19\)

This may lead them to select Choice B (\(-\mathrm{y}^2 + 19\))

The Bottom Line:

This problem tests the fundamental skill of correctly applying the distributive property with negative signs. The key insight is that the negative sign in front of parentheses must be distributed to every term inside, and students often forget to apply it to variable terms.