X2-x3+2x2+x+1 by X-1 Synthetic method

We want to divide the polynomial $x^3 - x^2 + 2x^2 + x + 1$ by $x-1$. First, simplify the polynomial: $x^3 + x^2 + x + 1$.

Using synthetic division with the root 1 from the divisor $x-1$:

1 | 1   1   1   1
  |     1   2   3
  ----------------
    1   2   3   4

The coefficients of the quotient are $1, 2, 3$, and the remainder is 4.

The quotient is $x^2 + 2x + 3$ and the remainder is 4.

So, $\frac{x^3 + x^2 + x + 1}{x-1} = x^2 + 2x + 3 + \frac{4}{x-1}$.

The final answer is $x^2 + 2x + 3 + \frac{4}{x-1}$.