(Case of double roots)

Let $P \in \mathbb{R}[X]$ and $a \in \mathbb{R}$. Determine the remainder of the Euclidean division of $P$ by $(X-a)^{2}$ in terms of $P(a)$ and $P^{\prime}(a)$.
Same principle as in the previous example, with the nuance that it will be necessary to derive the relation, then re-evaluate it at $a$ to obtain a second equation. In the end, we find that the remainder of the Euclidean division is $R=P(a)+P^{\prime}(a)(X-a)$.