[tex]\displaystyle n^4-2n^3+2n^2-2n+1= \\ \\ =(n^4-2n^3+n^2)+(n^2-2n+1)= \\ \\ =n^2(n-1)^2+(n-1)^2= \\ \\ =(n-1)^2(n^2+1) \\ \\ n^3-n^2+n-1=n^2(n-1)+(n-1)=(n-1)(n^2+1). \\ \\ Deci~ \frac{n^4-2n^3+2n^2-2n+1}{n^3-n^2+n-1}= \frac{(n-1)^2(n^2+1)}{(n-1)(n^2+1)}=n-1 \in \mathbb{N}. [/tex]
