Which expression is equivalent to \(-8(3\mathrm{y} + 5)\)?-{24y - 40}-{24y - 13}-{24y + 40}-{24y + 5}

1. INFER the mathematical approach needed

• Given: \(-8(3y + 5)\)
• What this tells us: We need to distribute the -8 to both terms inside the parentheses
• Strategy: Use the distributive property \(a(b + c) = ab + ac\)

2. SIMPLIFY by applying the distributive property

• Apply: \(-8(3y + 5) = -8(3y) + (-8)(5)\)
• This breaks our problem into two separate multiplications

3. SIMPLIFY each multiplication

• First term: \(-8(3y) = -24y\)
• Second term: \((-8)(5) = -40\)
• Combined: \(-24y + (-40)\)

4. SIMPLIFY the final expression

• \(-24y + (-40) = -24y - 40\)

Answer: A (\(-24y - 40\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up the distributive property but make a sign error when multiplying \(-8 \times 5\), getting +40 instead of -40.

Their work looks like: \(-8(3y + 5) = -24y + 40\)

This leads them to select Choice C (\(-24y + 40\))

Second Most Common Error:

Incomplete INFER reasoning: Students recognize they need to distribute but only apply it to the first term, treating the problem as \(-8(3y) + 5\) instead of \(-8(3y + 5)\).

Their reasoning: "I'll multiply -8 times 3y and then just add the 5"

This causes them to select Choice D (\(-24y + 5\))

The Bottom Line:

The distributive property requires careful attention to both the multiplication of each term AND the proper handling of negative signs throughout the entire process.