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

1. Divide the first term of the dividend \(2x^2\) by the first term of the divisor \(x\):

\(\frac{2x^2}{x} = 2x\)

2. Multiply the entire divisor \(x + 2\) by \(2x\):

\(2x(x + 2) = 2x^2 + 4x\)

3. Subtract \(2x^2 + 4x\) from \(2x^2\):

\(2x^2 - (2x^2 + 4x) = -4x\)

4. Divide \(-4x\) by \(x\):

\(\frac{-4x}{x} = -4\)

5. Multiply the entire divisor \(x + 2\) by \(-4\):

\(-4(x + 2) = -4x - 8\)

6. Subtract \(-4x - 8\) from \(-4x\):

\(-4x - (-4x - 8) = 8\)

The quotient is \(2x - 4\) and the remainder is 8.