unknown polynomial divided by $x^2(x-1)$, find the remainder.

We have

$$f(x)=(x-1)^2q_1(x)+x+3$$

$$f'(x)=2(x-1)q_1(x)+(x-1)^2q_1'(x)+1$$

Where for $x=1$ we have $f(1)=4$ and $f'(1)=1$

Then from

$$f(x)=x^2q_2(x)+2x+4$$ $$f'(x)=2xq_2(x)+x^2q_2'(x)+2$$

Where for $x=0$ we have $f(0)=4$ and $f'(0)=2$

Now from $$f(x)=x^2(x-1)q_3(x)+ax^2+bx+c$$ and

$$f'(x)=x^2q_3(x)+2x(x-1)q_3(x)+(x-1)x^2q_3'(x)+2ax+b$$

When we substitute the values $x=0$ and $x=1$ in $f$ and $f'$ we get

$f(0)=c=4$ and $f(1)=a+b+4=4$ $$a+b=0$$

$f'(0)=b=2$ from this we have $a=-2$. Thus the remainder is $r(x)=-2x^2+2x+4$