The given matrix equation is:

$\begin{bmatrix} 4 \\ 1 \end{bmatrix} X = \begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}$

To solve this, we need to recognize that this is a system of equations. The vector $\begin{bmatrix} 4 \\ 1 \end{bmatrix}$ represents the unknown variables multiplied by the matrix on the right-hand side. This means the equation can be written as two simultaneous equations.

1. First row of the matrix: $4x_1 + 1x_2 = -4$
2. Second row of the matrix: $4x_1 + 1x_2 = 8$

We can solve these equations by standard methods of solving a system of