Which expression is equivalent to \(8(\mathrm{x}^2 + 3) - 24\)?8x^2 + 248x^2 - 218x^28x^2 + 48

1. INFER the approach

• Given: \(8(\mathrm{x}^2 + 3) - 24\)
• Strategy: Apply distributive property first, then combine any like terms
• The parentheses indicate multiplication should happen before subtraction

2. SIMPLIFY using the distributive property

• Multiply 8 by each term inside the parentheses:
  • \(8 \times \mathrm{x}^2 = 8\mathrm{x}^2\)
  • \(8 \times 3 = 24\)
• So \(8(\mathrm{x}^2 + 3)\) becomes \(8\mathrm{x}^2 + 24\)

3. SIMPLIFY by substituting back into the original expression

• Replace \(8(\mathrm{x}^2 + 3)\) with \(8\mathrm{x}^2 + 24\):
• Original: \(8(\mathrm{x}^2 + 3) - 24\)
• Now: \((8\mathrm{x}^2 + 24) - 24\)

4. SIMPLIFY by combining the constant terms

• Combine: \(+24 - 24 = 0\)
• Final result: \(8\mathrm{x}^2 + 0 = 8\mathrm{x}^2\)

Answer: C (\(8\mathrm{x}^2\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly apply the distributive property by only multiplying the first term.

They calculate \(8(\mathrm{x}^2 + 3)\) as \(8\mathrm{x}^2 + 3\) instead of \(8\mathrm{x}^2 + 24\). Then they get:

\(8\mathrm{x}^2 + 3 - 24 = 8\mathrm{x}^2 - 21\)

This may lead them to select Choice B (\(8\mathrm{x}^2 - 21\))

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly apply the distributive property but forget to complete the subtraction step.

They correctly get \(8(\mathrm{x}^2 + 3) = 8\mathrm{x}^2 + 24\), but then stop there without subtracting 24 from the original expression.

This may lead them to select Choice A (\(8\mathrm{x}^2 + 24\))

The Bottom Line:

This problem tests systematic algebraic manipulation through multiple steps. Students must both apply the distributive property correctly AND remember to complete all operations in the original expression.