We are given the vector equation:

$x \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix} + y \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix} + z \begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} + 3 \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix} + 4$

Step 1: Simplify the right-hand side

The right-hand side consists of two vector terms plus a scalar value 4. First, calculate the sum of the vectors:

$\begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} + 3 \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} + \begin{pmatrix} 3 \\ -3 \\ 3 \end{pmatrix} = \begin{pmatrix} 6 \\ -5 \\ 4 \end{pmatrix}$

Next, we add the scalar 4 to each component of the resulting vector:

$\begin{pmatrix} 6 \\ -5 \\ 4 \end{pmatrix} + 4 = \begin{pmatrix} 10 \\ -1 \\ 8 \end{pmatrix}$

So the simplified right-hand side is:

$\begin{pmatrix} 10 \\ -1 \\ 8 \end{pmatrix}$

Step 2: Set up the vector equation

Now we have the equation:

$x \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix} + y \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix} + z \begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix} = \begin{pmatrix} 10 \\ -1 \\ 8 \end{pmatrix}$

We can now break this into three separate equations based on the components:

1. $-2x + 2y + z = 10$
2. $3x - 2y + z = -1$
3. $-x + y - 2z = 8$

Step 3: Solve the system of equations

We have the system:

1. $-2x + 2y + z = 10$
2. $3x - 2y + z = -1$
3. $-x + y - 2z = 8$

We can solve this using substitution or elimination. Let’s solve it step by step.

Step 3.1: Solve for $z$ from equation 1

From equation (1), solve for $z$:

z = 10 + 2x - 2y \tag{4}

Step 3.2: Substitute $z$ into equation (2)

Substitute $z = 10 + 2x - 2y$ into equation (2):

$3x - 2y + (10 + 2x - 2y) = -1$ Simplify:

$3x - 2y + 10 + 2x - 2y = -1$ $5x - 4y + 10 = -1$ 5x - 4y = -11 \tag{5}

Step 3.3: Substitute $z$ into equation (3)

Substitute $z = 10 + 2x - 2y$ into equation (3):

$-x + y - 2(10 + 2x - 2y) = 8$ Simplify:

$-x + y - 20 - 4x + 4y = 8$ $-5x + 5y = 28$ $x - y = -\frac{28}{5} \quad (\text{Divide by } -5)$