Consider the differential equation

$$(E): \quad y' + 10y = \left(30x^2 + 22x - 8\right)\mathrm{e}^{-5x+1} \quad \text{with} \quad x \in \mathbb{R}$$

where $y$ is a function defined and differentiable on $\mathbb{R}$.

\begin{enumerate}
  \item Solve on $\mathbb{R}$ the differential equation: $y' + 10y = 0$.
  \item Let the function $f$ defined on $\mathbb{R}$ by
$$f(x) = \left(6x^2 + 2x - 2\right)\mathrm{e}^{-5x+1}$$
We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ the derivative function of the function $f$. Justify that $f$ is a particular solution of $(E)$.
  \item Give the expression of all solutions of $(E)$.
\end{enumerate}