<s>[INST] Find the remainder when $f(x)$ divided by $(x^2 + x + 1)(x+1)$. When a polynomial $f(x)$ is divided by $x^2 + x + 1$ and $(x+1)^2$, the remainder are $x+5$ and $x-1$ respectively. Find the remainder when $f(x)$ is divided by $(x^2 + x + 1)(x+1)$.
First, I let the remainder be $Ax^2 + Bx +C$, then I try to find the values of $A$, $B$ and $C$. There is $3$ unknowns so we need three equations but I can only get two equations. 
 [/INST] Using the Generalized Euclidean Algorithm, we get
$$
(x+2)\left(x^2+x+1\right)-(x+1)(x+1)^2=1\tag1
$$
Therefore, we have
$$
\begin{align}
(x+2)\left(x^2+x+1\right)&\equiv1&&\pmod{(x+1)^2}\\
(x+2)\left(x^2+x+1\right)&\equiv0&&\pmod{x^2+x+1}
\end{align}\tag2
$$
and
$$
\begin{align}
-(x+1)(x+1)^2&\equiv0&&\pmod{(x+1)^2}\\
-(x+1)(x+1)^2&\equiv1&&\pmod{x^2+x+1}
\end{align}\tag3
$$
Thus, to get
$$
\begin{align}
f(x)&\equiv x-1&&\pmod{(x+1)^2}\\
f(x)&\equiv x+5&&\pmod{x^2+x+1}
\end{align}\tag4
$$
we can use $x-1$ times $(2)$ plus $x+5$ times $(3)$ to get $\bmod{(x+1)^2\left(x^2+2x+1\right)}$
$$
\begin{align}
f(x)
&\equiv(x-1)(x+2)\left(x^2+x+1\right)-(x+5)(x+1)(x+1)^2\\
&=-6x^3-18x^2-17x-7\\
&\equiv-6x^2-5x-1\pmod{(x+1)\left(x^2+2x+1\right)}\tag5
\end{align}
$$
</s>