You can use the polynomial long division method.
Stage 1
When you divide $x^4$ by $x$, you get $x^3$. When you multiply $x^3$ by $x - 1$, you get $x^4 - x^3$. After subtracting that from $x^4 - 1$, you get $x^3 - 1$. From there, you have $x^3 + \frac {x^3 - 1}{x - 1}$
Stage 2
You can apply the same method and get: $x^2 + \frac {x^2 - 1}{x - 1}$ After this stage, you get: $x^3 + x^2 + \frac{x^2 - 1}{x - 1}$ You will begin to notice a pattern. The degree gets smaller when you have to divide in every step.
Stage 3
Following the pattern, when you divide $x^2 - 1$ by $x - 1$, you get $x + \frac {x - 1}{x - 1}$ At the end of this stage, you get $x^3 + x^2 + x + \frac {x - 1}{x - 1}$
Stage 4
Following the pattern (in which this stage will be easy), you can use the $\frac aa = 1$ formula and get 1.
End of Problem
At the end, you will have $x^3 + x^2 + x + 1$. Hmm, something's weird here. In every stage you wen through, the degree got smaller. Also in the answer, the degree gets smaller! Anyways, the answer is $x^3 + x^2 + x + 1$.