\textbf{Part 1}

We consider the function $f$ defined on the set of real numbers $\mathbb{R}$ by:
$$f(x) = \left(x^2 - 4\right)\mathrm{e}^{-x}$$
We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function.

\begin{enumerate}
  \item Determine the limits of the function $f$ at $-\infty$ and at $+\infty$.
  \item Justify that for all real $x$, $f'(x) = \left(-x^2 + 2x + 4\right)\mathrm{e}^{-x}$.
  \item Deduce the variations of the function $f$ on $\mathbb{R}$.
\end{enumerate}

\textbf{Part 2}

We consider the sequence $(I_n)$ defined for all natural integer $n$ by $I_n = \int_{-2}^{0} x^n \mathrm{e}^{-x}\,\mathrm{d}x$.

\begin{enumerate}
  \item Justify that $I_0 = \mathrm{e}^2 - 1$.
  \item Using integration by parts, demonstrate the equality:
$$I_{n+1} = (-2)^{n+1}\mathrm{e}^2 + (n+1)I_n$$
  \item Deduce the exact values of $I_1$ and $I_2$.
\end{enumerate}

\textbf{Part 3}

\begin{enumerate}
  \item Determine the sign on $\mathbb{R}$ of the function $f$ defined in Part 1.
  \item The curve $\mathscr{C}_f$ of the function $f$ is represented in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. The domain $D$ of the shaded region is bounded by the curve $\mathscr{C}_f$, the $x$-axis and the $y$-axis. Calculate the exact value, in square units, of the area $S$ of the domain $D$.
\end{enumerate}