Some of your plus and minus signs didn't come out in the question, so I had to guess on them. The answer may not be completely correct for that reason (in fact, I suspect it's not because of the large remainder). However, I did my best to explain each step for you, so you know how to do this if my choice of signs was incorrect. Feel free to message me if you want more help.
x^2 - x - 34 Remainder 96/(x-5)
x - 5 x^3 - 4x^2 - 39x + 24 <--- Line 1
- (x^3 -5x^2) <--- Line 2
0 + x^2 - 39x +24 <--- Line 3
- (x^2 - 5x) <---- Line 4
0 + -34x + 24 <---- Line 5
- (-34x + 120) <---- Line 6
0 + -96 <---- Line 7
1) x goes into x^3: x^2 times. To get the first number under the division sign, you multiply x^2 by (x-5) - giving x^3 - 5x^2. (Line 2)
2) Subtract (x^3-5x^2) from the x^3 - 4x^2 - 39x + 24, giving x^2 - 39x + 24 (Line 3)
3) x goes into x^2: x times. Multiply x by (x-5) - giving x^2 - 5x (Line 4).