Which expression is equivalent to 8 + d^2 + 3?

1. TRANSLATE the problem information

• Given: \(8 + \mathrm{d}^2 + 3\)
• Find: Which expression is equivalent to this

2. INFER the approach needed

• I need to simplify this expression by combining like terms
• The constants 8 and 3 can be combined since they're both constant terms
• The \(\mathrm{d}^2\) term stays separate since it's not a constant

3. SIMPLIFY by combining like terms

• Rearrange for clarity: \(\mathrm{d}^2 + 8 + 3\)
• Add the constants: \(8 + 3 = 11\)
• Final simplified form: \(\mathrm{d}^2 + 11\)

Answer: B (\(\mathrm{d}^2 + 11\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make basic arithmetic errors when adding \(8 + 3\), perhaps getting 5, 24, or other incorrect sums.

If they calculate \(8 + 3 = 5\), this leads them to \(\mathrm{d}^2 + 5\), making them select Choice C (\(\mathrm{d}^2 + 5\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misread the operation signs in the original expression, seeing subtraction where there's addition.

If they misread \(8 + \mathrm{d}^2 + 3\) as involving subtraction, they might work with incorrect terms and end up confused, leading to guessing among the remaining choices.

The Bottom Line:

This problem tests fundamental algebra skills - recognizing like terms and performing basic arithmetic accurately. Even though it looks simple, small errors in reading or calculation can easily lead to wrong answers.