Polynomial division of $\frac{x^5+1}{x^4 + x^3 + x^2}$

Question

Task:

Do polynomial division for

$$\frac{x^5+1}{x^4 + x^3 + x^2}$$

The question is really about finding all primitive functions via partial fractions, but I know how to do all the other steps without a problem once I have it in a form that I can use partial fraction reasoning on.

Attempted solution:

Because the exponent is higher for the numerator, we can use polynomial division:

 x - 1 + 1/ x^2 - 1/ x^3
 -----------------------------------
 x^5 + 1              | x^4 +x^3 + x^2
                      _____________
-(x^5+x^4+x^3)
----------------------
 -x^4 - x^3 + 1
-(-x^4-x^3 - x^2)
----------------------
 x^2 + 1
 -(x^2 + x + 1)
-----------------------
 -x
-(-x - 1 - 1/x)
-----------------------
1 + 1/x

This just seems to go on and on and on without reaching any meaningful end. It also seems possible to factor out $x^2$ from the denominator and maybe something from the numerator, but I cannot find a way to make them cancel out in a meaningful way. Perhaps forcibly factor out $x^2$ from the numerator?

Any advice on how to proceed with this question?

Answer 1

$\frac{x^5+1}{x^4 + x^3 + x^2}$

Using long division (we stop when we get to power less than 4):

Answer 2

Note that$$x^5+1=(x^4+x^3+x^2)\times x-x^4-x^3+1$$and that$$-x^4-x^3+1=(x^4+x^3+x^2)\times(-1)+x^2+1.$$Therefore,$$x^5+1=(x^4+x^3+x^2)\times(x-1)+x^2+1$$and so$$\frac{x^5+1}{x^4+x^3+x^2}=x-1+\frac{x^2+1}{x^4+x^3+x^2}.$$