Find the remainder of x^2-3 when divided by x+1.

You can do this by either synthetic division or long division. I choose synthetic because it's easier. Remember, you need the root to do synthetic division, so for the function, the root is at x = -1. Let's set it up:

-1 | 1     0     -3
    
            -1      1
      1     -1    -2

These are the coefficients of our depressed polynomial. The polynomial will have 1 less degree in the answer, and it will look like:


As you can see, the remainder by doing that is -2, but in polynomial form you would write it as: 2/x+1. Hopefully, this makes sense to you!