Verified answer

We have to divide the polynomials until the degree of remainder in the process of division not become less than degree of denominator.

Let us expand (x-1)^3 the denominator first. We get

(x-1)^3=x^3-3x^2+3x-1

Now multiply x^3-3x^2+3x-1 by x we get x^4-3x^3+3x^2-x.

First part of the quotient is x and remaining polynomial is (x^4+x)-(x^4-3x^3+3x^2-x)=3x^3-3x^2+2x.

Now multiply x^3-3x^2+3x-1 by 3 we get 3x^3-9x^2+9x-3.

Now quotient becomes x+3 and remaining polynomial is

(3x^3-3x^2+2x)-(3x^3-9x^2+9x-3)=6x^2-7x+3

Now the degree of remainder (6x^2-7x+3) is 2 which is less than degree of denominator (x^3-3x^2+3x-1, which is 3) so the answer is

(x+3) with remainder (6x^2-7x+3)