What is the remainder when (x^3 + 5x^2 + 2x - 8) is divided by (x^2 + 2x)?

To determine the remainder when dividing the polynomial (x^3 + 5x^2 + 2x - 8) by (x^2 + 2x), we can utilize polynomial long division or synthetic division methods. When dividing, the degree of the remainder must be less than the degree of the divisor. Since the divisor (x^2 + 2x) is a degree 2 polynomial, the remainder will be a linear polynomial of the form (Ax + B).

Starting with the polynomial (x^3 + 5x^2 + 2x - 8), we divide by (x^2 + 2x):

1. Divide (x^3) by (x^2) to get (x).

2. Multiply (x) by the entire divisor (x^2 + 2x) to get (x^3 + 2x^2).

3. Subtract this from (x^3 + 5x^2 + 2x - 8) yielding (3x^2 + 2x - 8).

4. Next, divide (3x^2) by (