Let me try this:

For:

4x (x^2 + 2) = 4x^3 + 8x

Is it ok to append a zero to get the answer?

As follows:

4x^3 + x^2 + 8x + 4

4x^3 + 0 + 8x

__________________

x^2 + 4

Next Step:

1 (x^2 + 2) = x^2 + 2

x^2 + 4

x^2 + 2

_______

2

This approach of adding a zero to the long division allows me to get 4x+1 remainder 2. Is that right?