5x3+x2-3x+7 by x-1
Use either long division or synthetic division to divide
Synthetic is easier.
1 | 5 1 -3 7
5 6 3
5 6 3 10
Thus: 5x^2 + 6x + 3 + 10/(x-1)
I'll guide you through the long division method. It does help if you remember how to do long division of integers.
Start by trying to divide x-1 into 5x3+x2. Focus on what you would need to multiply x by to get 5x3. If you multiply x by 5x2 you get 5x3, so let's try (x-1)(5x2) which gives 5x3-5x2.
Write 5x3-5x2 directly underneath 5x3+x2 and subtact. You get 6x2 because x2-(-5x2) = 6x2
So the first term in your answer is 5x2 and you have a remainder of 6x2. Bring down the next term in your dividend, which is -3x. Add it to your remainder and start again.
This time you are dividing x-1 into 6x2-3x. Focus on the x and the 6x. Keep doing this same process until you have brought everything down and divided. You will either have no remainder at the end, which means it divided evenly, or you will have something left. If there is a remainder, take that expression and divide it by x-1, your divisor. That fraction is added on to the end of your quotient.