Calculate the quotient and the remainder of the Euclidean division of $X^{3}+2 X^{2}+3 X+4$ by $X^{2}+1$.

## Definition 9.

Let $A$ and $B$ be two polynomials in $\mathbb{K}[X]$. We say that $B$ divides $A$ if there exists a polynomial $Q$ in $\mathbb{K}[X]$ such that $A=B Q$.

In other words, $B$ divides $A$ if the remainder of the Euclidean division of $A$ by $B$ is zero.
The method used to prove Theorem 7 allows us, given a pair $(Q, R)$ of polynomials for which $A = B Q + R$ and $\operatorname{deg}(R) \geqslant \operatorname{deg}(B)$, to compute a new pair $\left(Q_{1}, R_{1}\right)$ for which $A = B Q_{1} + R_{1}$ and $\operatorname{deg}\left(R_{1}\right) < \operatorname{deg}(R)$. By proceeding in this manner step by step, we will eventually identify the sought quotient and remainder.

Here, we successively encounter the pairs

$$
\begin{aligned}
& \left(Q_{0}, R_{0}\right) = \left(0, X^{3} + 2 X^{2} + 3 X + 4\right) \\
& \left(Q_{1}, R_{1}\right) = \left(X, 2 X^{2} + 2 X + 4\right) \\
& \left(Q_{2}, R_{2}\right) = (X + 2, 2 X + 2)
\end{aligned}
$$

Since $\operatorname{deg}\left(R_{2}\right) = 1 < \operatorname{deg}\left(X^{2} + 1\right)$, we stop here: the sought quotient and remainder are therefore $Q_{2}(X) = X + 2$ and $R_{2}(X) = 2 X + 2$.