Is x-2 a factor of 2x3+x2-3

No ... if it was 2x^3+x^2-3 would equal zero when x=2. You could divide and demonstrate for yourself that the division would have a remainder ...

2x^3+x^2-3 / x-2

2x^2 rem 5x^2-3

5x rem 10x-3

10 rem 17 so if we factor out (x-2) we have:

(x-2) (2x^2+5x+10) + 17 / (x-2)

We have the remainder 17 / (x-2), if (x-2) was a factor the remainder would equal zero.

No the factor of 2x3+x2-3 is 3+2x