If b = r - s, which of the following is equivalent to 4x^2 - r^2 + 2rs - s^2?

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{b = r - s}\) (substitution relationship)
  • Expression to simplify: \(\mathrm{4x^2 - r^2 + 2rs - s^2}\)
• We need to rewrite this expression in terms that match one of the answer choices.

2. INFER the approach

• The key insight is to look for patterns in the r and s terms that might simplify when we use the given relationship \(\mathrm{b = r - s}\).
• Since we have multiple terms with r and s, let's group them together to see if they form a recognizable pattern.

3. SIMPLIFY by grouping and factoring

• Group the r and s terms: \(\mathrm{4x^2 + (-r^2 + 2rs - s^2)}\)
• Notice that \(\mathrm{-r^2 + 2rs - s^2}\) looks like it might be factorable.
• Factor out the negative: \(\mathrm{-r^2 + 2rs - s^2 = -(r^2 - 2rs + s^2)}\)
• Recognize the perfect square trinomial: \(\mathrm{r^2 - 2rs + s^2 = (r - s)^2}\)
• So we have: \(\mathrm{4x^2 - (r - s)^2}\)

4. INFER the substitution opportunity

• Since \(\mathrm{b = r - s}\), we can substitute: \(\mathrm{4x^2 - (r - s)^2 = 4x^2 - b^2}\)

5. INFER the final pattern recognition

• Now we have \(\mathrm{4x^2 - b^2}\), which is a difference of squares pattern.
• Recall that \(\mathrm{a^2 - b^2 = (a + b)(a - b)}\)
• Here, \(\mathrm{4x^2 = (2x)^2}\), so: \(\mathrm{4x^2 - b^2 = (2x + b)(2x - b)}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(\mathrm{-r^2 + 2rs - s^2}\) forms a perfect square trinomial pattern.

Students see the scattered r and s terms and either try to substitute \(\mathrm{b = r - s}\) directly into individual terms, or they attempt to expand the answer choices to match. Without recognizing the perfect square trinomial structure, they can't make progress toward the elegant solution. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic mistakes when factoring or applying the difference of squares.

Even students who recognize the perfect square trinomial might write \(\mathrm{r^2 - 2rs + s^2 = (r + s)^2}\) instead of \(\mathrm{(r - s)^2}\), or they might incorrectly apply the difference of squares formula. These calculation errors can lead them to select Choice A (\(\mathrm{(2x + b)^2}\)) or Choice B (\(\mathrm{(2x - b)^2}\)) if they think the expression equals a perfect square instead of a product of factors.

The Bottom Line:

This problem requires students to see algebraic patterns within a jumble of terms and make strategic grouping decisions. Success depends on recognizing two classic patterns: perfect square trinomials and difference of squares. Students who memorized these formulas but can't spot when to apply them will struggle significantly.