Did i correctly solve this problem using synthetic division

(2x^2 -5x +3)/(x -3)
What I did: divided by x=3
2x + 1 with a remainder of 6
So my answers are: x=3 and x= -.5

Use coefficients of polynomial only: 2, -5, 3

Synthetic division:
3 | 2 .. -5 ... 3
.. | ...... 6 ... 3
.. ----------------
.... 2 ... 1 ... 6

So you are correct. But x = 3 and x = -0.5 does not really make sense in this context, since you are not asked to solve an equation. Also, x = 3 is not a root of original quadratic equation, since remainder when dividing by (x - 3) is not 0. What results do tell us is:

(2x^2 - 5x + 3) / (x - 3) = (2x + 1) + 6 / (x - 3)

or

2x^2 - 5x + 3 = (2x + 1) (x - 3) + 6

You divided correctly but I have no idea what those x values represent.

the final answer should be 2x+1+6/x-3