Divide the following expression using long division. (x3 + x2 + x – 3) ÷ (x – 1)?

As with any long division problem, you start with the most significant terms of the dividend and divisor.   x^3/x = x^2
This is the first term of the quotient.
As with any long division problem, you multiply this by the divisor, subtract from the dividend and repeat.
  (x^3 + x^2 + x - 3) - x^2(x-1)
  = 2x^2 + x - 3
The second term of the quotient will be
  2x^2/x = 2x
The new dividend is
  (2x^2 + x - 3) - 2x(x-1)
  = 3x - 3
We can see that this is 3(x-1), so the remaining term of the quotient is 3.

The quotient is x^2 + 2x + 3.