The bottom part, 3x + 2, set it equal to 0:
3x + 2 = 0
3x = -2
x = -2/3
If 3x + 2 can be factored out of 2x^3 + 2x^2 - 4x + 8, then plugging x = -2/3 into the top should make the top 0. Let's see:
2(-2/3)^3 + 2(-2/3)^2 -4(-2/3) + 8
2(-8/27) + 2(4/9) + 8/3 + 8
-16/27 + 8/9 + 8/3 + 8
-16/27 + 24/27 + 72/27 + 202/27
282/27
The result is not 0, so 3x + 2 does not factor out of 2x^3 + 2x^2 - 4x + 8.
You can also look at the graph of 2x^3 + 2x^2 - 4x + 8 here ( https://www.google.com/?gws_rd=ssl#q=plot(2x%5E3+%2B+2x%5E2+-+4x+%2B+8) ) and see that it doesn't have a zero at -2/3.
Since the bottom does not factor out of the top, the quotient of (2x^3 + 2x^2 - 4x + 8)/(3x+2) should just be (2x^3 + 2x^2 - 4x + 8)/(3x+2).