Find the remainder when (x3 + 4x2 + 6x − 2) is divided by (x + 5).

Concept: The Remainder Theorem states that when a polynomial p(x) is divided by a linear factor (x − a), the remainder equals p(a) — the value obtained by substituting x = a directly into the polynomial, without carrying out long division.

Application: Applying this to the given polynomial:

1. Write the divisor (x + 5) in the form (x − a): here a = −5.

2. Substitute x = −5 into p(x) = x³ + 4x² + 6x − 2: p(−5) = (−5)³ + 4(−5)² + 6(−5) − 2.

3. Evaluate each term: (−5)³ = −125, 4(−5)² = 4 × 25 = 100, and 6(−5) = −30.

4. Add the four terms: −125 + 100 − 30 − 2 = −57.

Cross-check: Dividing x³ + 4x² + 6x − 2 by (x + 5) using long division gives quotient x² − x + 11 and remainder −57, matching the value found by direct substitution.

So the remainder is −57.