How do i divide this? f(x) / g(x) f(x)= 3x^2 + 2x - 4 and g(x)= x - 2 also how do i use the remainder theorem to check that the remainder obtained by synthetic division is correct?

Use synthetic division:

2 | 3 2 -4
   |    6  16      
   
     3 8  12

So f(x)/g(x) = 3x + 8 + 12/(x-2)
Where the remainder is: 12/(x-2)

The remainder theorem tells us that if you divide a polynomial by a linear factor in the form x - a, then the remainder is just f(a)

So if we do f(2), the remainder tells us that we should get 12.  So in order to check if our synthetic division is correct, we must check if f(2) = 12

It does equal 12, so therefore our work is correct.