Answer:
x^2-6x+7
Explanation:
You can use synthetic division since you are dividing by a linear factor with leading coefficient 1. (You can also do synthetic division when dividing by a linear factor whose leading coefficient is not 1 but it is a bit trickier.)
So since we are dividing by x-2, 2 goes on the outside.
(If we were dividing by x+2, -2 would go on the outside.)
Now the thing that goes on top inside is the dividend, the numerator which is 1x^3-8x^2+19x-14. If you were missing any terms you would have to place a zero but we aren't. All exponents between 3 and 0 (inclusive meaning to include the 3 and the constant term) on variable x is accounted for.
2| 1 -8 19 -14
| 2 -12 14
-----------------------------
1 -6 7 0
So the remainder is 0 and the quotient is x^2-6x+7.