_________________________
x^2-3x+4 / 2x^3+2x^2-16x+32
Note that x^2 divides into 2x^3 with a partial quotient of 2x. Multiply x^2-3x+4 by this 2x and write your product under 2x^3+2x^2-16x+32:
__2x _________________________
x^2-3x+4 / 2x^3+2x^2-16x+32
2x^3 -6x^2 + 8x
----------------------- subtract
8x^2 -24x + 32
Now divide x^2 into 8x^2: result is 8.
__2x + 8_________________________
x^2-3x+4 / 2x^3+2x^2-16x+32
2x^3 -6x^2 + 8x
-----------------------
8x^2 -24x + 32
8x^2 -24x + 32
------------------------ subtract
0
So the desired quotient is 2x+8, or 2(x+4).