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

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

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

3. Subtract \(2x^4 - 2x^3 + 4x^2\) from \(2x^4 + 1\) to get \(2x^3 - 4x^2 + 1\).

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

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

6. Subtract \(2x^3 - 2x^2 + 4x\) from \(2x^3 - 4x^2 + 1\) to get \(-2x^2 - 4x + 1\).

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

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

9. Subtract \(-2x^2 + 2x - 4\) from \(-2x^2 - 4x + 1\) to get \(-6x + 5\).

The quotient is \(2x^2 + 2x - 2\) and the remainder is \(-6x + 5\).