(a) Undefined for x=2 (because it would produce division by zero).
(b)
x-2 happens to be a factor of the numerator.
The quadratic factor can be obtained by synthetic division:
2 | 2 -3 -8 | 12
2 4 2 |-12
2 1 -6 | 0 = 2x²+x-6.
So (2x³-3x²-8x+12)/(x-2)=2x²+x-6=(2x-3)(x+2).
Zeroes are 3/2 and -2.