\((2\mathrm{x} + 5)^2 - (\mathrm{x} - 2) + 2(\mathrm{x} + 3)\)Which of the following is equivalent to the expression above?

1. INFER the approach

• This expression has three main parts: a binomial square, a linear expression with a negative sign, and a distributed term
• Strategy: Handle each part separately, then combine all terms

2. SIMPLIFY the binomial square \((2\mathrm{x} + 5)^2\)

• Use the pattern \((\mathrm{a} + \mathrm{b})^2 = \mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\)
• \((2\mathrm{x} + 5)^2 = (2\mathrm{x})^2 + 2(2\mathrm{x})(5) + (5)^2\)
• \((2\mathrm{x} + 5)^2 = 4\mathrm{x}^2 + 20\mathrm{x} + 25\)

3. SIMPLIFY the term \(-(\mathrm{x} - 2)\)

• Distribute the negative sign: \(-(\mathrm{x} - 2) = -\mathrm{x} + 2\)
• Key insight: The negative affects both terms inside the parentheses

4. SIMPLIFY the term \(2(\mathrm{x} + 3)\)

• Apply distributive property: \(2(\mathrm{x} + 3) = 2\mathrm{x} + 6\)

5. SIMPLIFY by combining all terms

• Substitute back: \(4\mathrm{x}^2 + 20\mathrm{x} + 25 + (-\mathrm{x} + 2) + (2\mathrm{x} + 6)\)
• Rewrite: \(4\mathrm{x}^2 + 20\mathrm{x} + 25 - \mathrm{x} + 2 + 2\mathrm{x} + 6\)
• Group like terms: \(4\mathrm{x}^2 + (20\mathrm{x} - \mathrm{x} + 2\mathrm{x}) + (25 + 2 + 6)\)
• Combine: \(4\mathrm{x}^2 + 21\mathrm{x} + 33\)

Answer: A. \(4\mathrm{x}^2 + 21\mathrm{x} + 33\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Incorrectly expanding \((2\mathrm{x} + 5)^2\) as \(4\mathrm{x}^2 + 25\), forgetting the middle term \(20\mathrm{x}\)

Students often remember that \((\mathrm{a} + \mathrm{b})^2\) involves squaring both terms but forget about the cross-product term \(2\mathrm{ab}\). They think \((2\mathrm{x} + 5)^2 = (2\mathrm{x})^2 + (5)^2 = 4\mathrm{x}^2 + 25\), missing the \(2(2\mathrm{x})(5) = 20\mathrm{x}\) term.

This leads them to get \(4\mathrm{x}^2 + 25 - \mathrm{x} + 2 + 2\mathrm{x} + 6 = 4\mathrm{x}^2 + \mathrm{x} + 33\), causing them to select Choice D (\(4\mathrm{x}^2 + \mathrm{x} + 33\)).

Second Most Common Error:

Poor SIMPLIFY execution: Mishandling the negative sign in \(-(\mathrm{x} - 2)\)

Students might distribute the negative incorrectly, treating \(-(\mathrm{x} - 2)\) as \(-\mathrm{x} - 2\) instead of \(-\mathrm{x} + 2\). This error occurs when students don't carefully apply the rule that a negative times a negative gives a positive.

This mistake leads to \(4\mathrm{x}^2 + 20\mathrm{x} + 25 - \mathrm{x} - 2 + 2\mathrm{x} + 6 = 4\mathrm{x}^2 + 21\mathrm{x} + 29\), causing them to select Choice B (\(4\mathrm{x}^2 + 21\mathrm{x} + 29\)).

The Bottom Line:

This problem tests systematic algebraic manipulation across multiple steps. Success requires careful attention to expanding binomial squares completely and correctly handling negative signs in distribution.