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

1. INFER the approach needed

• We have: \(\mathrm{4(3y^2) - 5y^2}\)
• Strategy: Apply distributive property first, then combine like terms
• Why this order? We must simplify \(\mathrm{4(3y^2)}\) before we can identify like terms

2. SIMPLIFY using the distributive property

• Apply distributive property to \(\mathrm{4(3y^2)}\):
  \(\mathrm{4(3y^2) = 4 \times 3y^2 = 12y^2}\)
• Expression now becomes: \(\mathrm{12y^2 - 5y^2}\)

3. SIMPLIFY by combining like terms

• Both terms have \(\mathrm{y^2}\), so they're like terms
• Combine coefficients: \(\mathrm{12y^2 - 5y^2 = (12 - 5)y^2 = 7y^2}\)

Answer: A (\(\mathrm{7y^2}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors in basic operations

• Error in multiplication: \(\mathrm{4 \times 3 = 11}\) instead of 12, leading to \(\mathrm{11y^2 - 5y^2 = 6y^2}\)
• Error in subtraction: \(\mathrm{12 - 5 = 8}\) instead of 7, leading to \(\mathrm{8y^2}\)

Neither of these incorrect results appears in the answer choices, which leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about order of operations: Students attempt to subtract \(\mathrm{5y^2}\) from \(\mathrm{3y^2}\) first

• They see \(\mathrm{4(3y^2 - 5y^2)}\) instead of \(\mathrm{4(3y^2) - 5y^2}\)
• This gives \(\mathrm{4(-2y^2) = -8y^2}\), which also doesn't match answer choices
• This leads to confusion and random answer selection

The Bottom Line:

This problem tests whether students can systematically apply the distributive property and combine like terms in the correct sequence. Success requires careful attention to both the algebraic procedures and basic arithmetic accuracy.