The task involves finding the intersection points between a line $g$ and a plane $E$. The parametric equation of the line $g$ is given as:

$\mathbf{x} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$

And the equation of the plane $E$ is provided in a parametric form:

$\mathbf{x} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + r \begin{pmatrix} 4 \\ -1 \\ -1 \end{pmatrix} + s \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}$

To find the intersection points, we need to set the parametric equation of the line equal to the parametric equation of the plane and solve for $t$, $r$, and $s$.

Substituting the equation of the line into the equation of the plane, we get:

$\begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + r \begin{pmatrix} 4 \\ -1 \\ -1 \end{pmatrix} + s \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}$

We will solve this system to find the values of $t$, $r$, and $s$. Let's start the calculations.