To divide \(x^4\) by \(x^2 + 2x - 1\), we use polynomial long division.

1. Divide the leading term \(x^4\) by \(x^2\) to get \(x^2\).

2. Multiply \(x^2\) by \(x^2 + 2x - 1\) to get \(x^4 + 2x^3 - x^2\).

3. Subtract \(x^4 + 2x^3 - x^2\) from \(x^4\) to get \(-2x^3 + x^2\).

4. Divide \(-2x^3\) by \(x^2\) to get \(-2x\).

5. Multiply \(-2x\) by \(x^2 + 2x - 1\) to get \(-2x^3 - 4x^2 + 2x\).

6. Subtract \(-2x^3 - 4x^2 + 2x\) from \(-2x^3 + x^2\) to get \(5x^2 - 2x\).

7. Divide \(5x^2\) by \(x^2\) to get \(5\).

8. Multiply \(5\) by \(x^2 + 2x - 1\) to get \(5x^2 + 10x - 5\).

9. Subtract \(5x^2 + 10x - 5\) from \(5x^2 - 2x\) to get \(-12x + 5\).

The quotient is \(x^2 - 2x + 5\) and the remainder is \(-12x + 5\).