To divide \(2x^4 - 27\) by \(x^2 + x + 3\), we perform polynomial long division:

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

2. Multiply \(2x^2\) by \(x^2 + x + 3\) to get \(2x^4 + 2x^3 + 6x^2\).

3. Subtract \(2x^4 + 2x^3 + 6x^2\) from \(2x^4 - 27\) to get \(-2x^3 - 6x^2 - 27\).

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

5. Multiply \(-2x\) by \(x^2 + x + 3\) to get \(-2x^3 - 2x^2 - 6x\).

6. Subtract \(-2x^3 - 2x^2 - 6x\) from \(-2x^3 - 6x^2 - 27\) to get \(-4x^2 - 6x - 27\).

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

8. Multiply \(-4\) by \(x^2 + x + 3\) to get \(-4x^2 - 4x - 12\).

9. Subtract \(-4x^2 - 4x - 12\) from \(-4x^2 - 6x - 27\) to get \(-2x - 15\).

The quotient is \(2x^2 - 2x - 4\) and the remainder is \(10x - 15\).