The greatest common factor is simply which factors of each number are shared between them. Factors are numbers that multiply together to get a certain number.
For example, a) 2x, 4
The factors of 2x are 2 and x
The factors of 4 are 2 and 2
The factors that are shared between them is 2, so the greatest common factor is 2 in this case.
For 2x^2, 4x it's easier to write it out:
The factors of 2x^2 are 2, x, x
The factors of 4x are 2, 2, x
Because they both have 2 and x, the greatest common factor is 2x in this case.