Question:\(3(\mathrm{x} - 1) + (2\mathrm{x} + 3)^2 - (\mathrm{x} + 4)\)Which of the following is equivalent to the expression above?

1. INFER the solution strategy

• Given: \(3(x - 1) + (2x + 3)^2 - (x + 4)\)
• Strategy: Expand each term separately, then combine like terms
• Order matters less than accuracy in expansion

2. SIMPLIFY by expanding each term

First term: \(3(x - 1)\)
Using distributive property: \(3(x - 1) = 3x - 3\)

Second term: \((2x + 3)^2\)
Using perfect square formula \((a + b)^2 = a^2 + 2ab + b^2\):

• \(a = 2x, b = 3\)
• \((2x + 3)^2 = (2x)^2 + 2(2x)(3) + (3)^2\)
• \((2x + 3)^2 = 4x^2 + 12x + 9\)

Third term: \(-(x + 4)\)
Distribute the negative: \(-(x + 4) = -x - 4\)

3. SIMPLIFY by combining all terms

Write out all terms: \(3x - 3 + 4x^2 + 12x + 9 - x - 4\)

4. SIMPLIFY by collecting like terms

• \(x^2\) terms: \(4x^2\)
• \(x\) terms: \(3x + 12x - x = 14x\)
• Constant terms: \(-3 + 9 - 4 = 2\)

Answer: \(4x^2 + 14x + 2\) (Choice B)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Incorrectly expanding \((2x + 3)^2\)

Many students forget the middle term in the perfect square expansion. They might calculate:
\((2x + 3)^2 = 4x^2 + 9\) (missing the \(12x\) term)

This leads to: \(3x - 3 + 4x^2 + 9 - x - 4 = 4x^2 + 2x + 2\)

This may lead them to select Choice A (\(4x^2 + 10x + 2\)) if they make an additional sign error, or get confused and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Sign errors when distributing the negative

Students might incorrectly handle \(-(x + 4)\) as \(-x + 4\) instead of \(-x - 4\).

With correct perfect square but wrong signs: \(3x - 3 + 4x^2 + 12x + 9 - x + 4 = 4x^2 + 14x + 10\)

This doesn't match any answer choice exactly, leading to confusion and guessing.

The Bottom Line:

This problem tests systematic algebraic manipulation. Success requires careful attention to both the perfect square expansion and sign management throughout the combining process.