The expression x^4 + 2x^2 - 15 can be written in the form \(\mathrm{(x^2 - a)(x^2 + b)}\), where a...

1. INFER the problem structure

• Given: \(\mathrm{x^4 + 2x^2 - 15}\) needs to be written as \(\mathrm{(x^2 - a)(x^2 + b)}\)
• Key insight: This expression is 'quadratic in form' because the powers of x are related (\(\mathrm{x^4 = (x^2)^2}\) and \(\mathrm{2x^2 = 2(x^2)}\))
• Strategy: Use substitution to convert this to a standard quadratic

2. SIMPLIFY using substitution

• Let \(\mathrm{u = x^2}\), so \(\mathrm{x^4 = u^2}\)
• The expression becomes: \(\mathrm{u^2 + 2u - 15}\)
• Now we have a standard quadratic to factor

3. SIMPLIFY the quadratic factoring

• Need two numbers that multiply to \(\mathrm{-15}\) and add to \(\mathrm{+2}\)
• Factor pairs of \(\mathrm{-15}\): \(\mathrm{(1, -15), (-1, 15), (3, -5), (-3, 5)}\)
• Check sums: \(\mathrm{(-3) + 5 = 2}\) ✓
• Therefore: \(\mathrm{u^2 + 2u - 15 = (u + 5)(u - 3)}\)

4. SIMPLIFY by substituting back

• Replace \(\mathrm{u}\) with \(\mathrm{x^2}\): \(\mathrm{(x^2 + 5)(x^2 - 3)}\)
• Reorder to match template form: \(\mathrm{(x^2 - 3)(x^2 + 5)}\)

5. INFER the final answer

• Compare \(\mathrm{(x^2 - 3)(x^2 + 5)}\) with \(\mathrm{(x^2 - a)(x^2 + b)}\)
• From \(\mathrm{(x^2 - 3)}\): \(\mathrm{a = 3}\)
• From \(\mathrm{(x^2 + 5)}\): \(\mathrm{b = 5}\)
• Both are positive constants as required

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(\mathrm{x^4 + 2x^2 - 15}\) is quadratic in form and attempting to factor it directly as a fourth-degree polynomial.

Students might try methods like grouping or looking for common factors, which don't work here. Without the key insight about substitution, they get stuck and either guess or select an answer based on partial work. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Correctly setting up \(\mathrm{u^2 + 2u - 15}\) but making errors when finding the factor pairs.

Students might incorrectly identify factor pairs (like using 3 and 5 instead of -3 and 5) or make sign errors when checking which pair adds to +2. This could lead them to get factors like \(\mathrm{(u + 3)(u - 5)}\), which would give \(\mathrm{a = 5}\) when substituted back and reordered, causing them to select Choice C (5).

The Bottom Line:

This problem tests whether students can recognize patterns in polynomial structure and apply the substitution technique—a key bridge between basic quadratic factoring and more advanced polynomial work.