The problem is asking you to write the vector $\begin{bmatrix} -4 \\ -2 \\ 2 \end{bmatrix}$ as a linear combination of three given vectors: $\begin{bmatrix} 0 \\ -2 \\ -4 \end{bmatrix}$, $\begin{bmatrix} -5 \\ 3 \\ 2 \end{bmatrix}$, and $\begin{bmatrix} 4 \\ 0 \\ 1 \end{bmatrix}$.

This can be done by solving the equation:

$\begin{bmatrix} -4 \\ -2 \\ 2 \end{bmatrix} = c_1 \begin{bmatrix} 0 \\ -2 \\ -4 \end{bmatrix} + c_2 \begin{bmatrix} -5 \\ 3 \\ 2 \end{bmatrix} + c_3 \begin{bmatrix} 4 \\ 0 \\ 1 \end{bmatrix}$

This is a system of three equations that we can solve for the unknown coefficients $c_1$, $c_2$, and $c_3$. Expanding the equation:

$\begin{bmatrix} -4 \\ -2 \\ 2 \end{bmatrix} = c_1 \begin{bmatrix} 0 \\ -2 \\ -4 \end{bmatrix} + c_2 \begin{bmatrix} -5 \\ 3 \\ 2 \end{bmatrix} + c_3 \begin{bmatrix} 4 \\ 0 \\ 1 \end{bmatrix}$

We have the system of equations:

$0c_1 - 5c_2 + 4c_3 = -4$ $-2c_1 + 3c_2 + 0c_3 = -2$ $-4c_1 + 2c_2 + 1c_3 = 2$

Now, solving this system of linear equations will give the values of $c_1$, $c_2$, and $c_3$.

Do you want me to solve this system for you, or would you like to try?

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Here are some related questions:

1. What is the significance of solving a system of linear equations in vector form?
2. How can you verify the solution once the coefficients are found?
3. Why is Gaussian elimination useful for solving such systems?
4. What is the geometrical interpretation of a linear combination of vectors?
5. How can matrix row reduction be used to solve more complicated linear systems?

Tip: When solving systems of equations, it is often useful to reduce the system using elimination methods to avoid errors when working with multiple variables.