Solve using synthetic division:
(x^5+2x^2+2x+1)/(x^2-1)

  x^3 +x +2 +3/(x -1)

Step-by-step explanation:

Synthetic division can be done using a divisor with a power of the variable other than one, but the most straightforward application of it uses a binomial divisor with a variable coefficient of 1. Fortunately the divisor in this problem is the difference of squares, so can be factored into two divisors. We can make use of synthetic division to divide by one of them, then we can use it again with the other divisor.

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

  = (x^4 -x^3 +x^2 +x +1)/(x -1)

  = x^3 +x +2 + 3/(x -1)

The remainder from the second division is simplified by the fact that there is no remainder from the first division. If desired, the final remainder can be expressed over the original divisor as (3x +3)/(x^2 -1) by multiplying numerator and denominator by the first divisor, (x+1).

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The synthetic division tableaux are attached. First, we divide by (x+1), then that result is divided by (x-1).

We assume you've been shown how this works. If not, there are numerous web sites and videos that can explain better than is possible in this space.

(x3 + 6x2 + 2x - 6) • (x - 1)

step-by-step explanation:

polynomial long division :

3.2     polynomial long division

dividing :   x4+5x3-4x2-8x+6

                              ("dividend")

by         :     x-1     ("divisor")

dividend     x4   +   5x3   -   4x2   -   8x   +   6

- divisor   * x3     x4   -   x3            

remainder         6x3   -   4x2   -   8x   +   6

- divisor   * 6x2         6x3   -   6x2        

remainder             2x2   -   8x   +   6

- divisor   * 2x1             2x2   -   2x    

remainder               -   6x   +   6

- divisor   * -6x0               -   6x   +   6

remainder                     0

quotient :   x3+6x2+2x-6   remainder:   0