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The given matrix equation can be expressed as:

\[
[x \quad 2 \quad 1] \cdot \begin{bmatrix} 4 & 4 & -4 \\ 4 & -1 & 0 \\ -4 & -18 & 2 \end{bmatrix} \cdot \begin{bmatrix} x \\ -1 \\ -4 \end{bmatrix} = [0]
\]

We need to compute this matrix multiplication to determine the value(s) of \( x \).

### Step 1: Multiply the first row vector by the 3x3 matrix.

Performing the first matrix multiplication:
\[
[x \quad 2 \quad 1] \cdot \begin{bmatrix} 4 & 4 & -4 \\ 4 & -1 & 0 \\ -4 & -18 & 2 \end{bmatrix}
\]

This gives the resulting row vector:
\[
[4x + 8 + -4, 4x - 2 + 0, -4x - 36 + 2]
\]
Simplifying each element:
\[
[4x + 4, 4x - 2, -4x - 34]
\]

### Step 2: Multiply the resulting row vector by the column vector.

Now, perform the multiplication of the row vector \([4x + 4, 4x - 2, -4x - 34]\) by the column vector \(\begin{bmatrix} x \\ -1 \\ -4 \end{bmatrix}\):
\[
(4x + 4)x + (4x - 2)(-1) + (-4x - 34)(-4)
\]

Now simplify each term:
\[
(4x + 4)x = 4x^2 + 4x
\]
\[
(4x - 2)(-1) = -4x + 2
\]
\[
(-4x - 34)(-4) = 16x + 136
\]

### Step 3: Combine all the terms.

Now, add all the terms together:
\[
4x^2 + 4x - 4x + 2 + 16x + 136 = 4x^2 + 16x + 138
\]

Set the equation equal to 0:
\[
4x^2 + 16x + 138 = 0
\]

### Step 4: Solve the quadratic equation.

We can divide the entire equation by 2 to simplify:
\[
2x^2 + 8x + 69 = 0
\]

Now, solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 2\), \(b = 8\), and \(c = 69\).

First, calculate the discriminant:
\[
\Delta = b^2 - 4ac = 8^2 - 4(2)(69) = 64 - 552 = -488
\]

Since the discriminant is negative, there are no real solutions. Therefore, the value of \(x\) is complex.

The solutions are:
\[
x = \frac{-8 \pm \sqrt{-488}}{4} = \frac{-8 \pm \sqrt{488}i}{4}
\]
Simplifying:
\[
x = \frac{-8 \pm 22.09i}{4}
\]
\[
x = -2 \pm 5.52i
\]

Thus, the solutions are:
\[
x = -2 + 5.52i, \quad x = -2 - 5.52i
\]

Would you like further details on any part of the solution?

### Follow-up questions:
1. How do you simplify the quadratic equation further?
2. What is the significance of a negative discriminant in a quadratic equation?
3. How can you confirm if your solution is correct for complex numbers?
4. What are the properties of complex roots in quadratic equations?
5. Could there be a scenario where \(x\) has real values for this kind of matrix equation?

**Tip**: Always check for real solutions before calculating the complex roots in quadratic equations.

Determine the value(s) of x such that [x  2  1] [4  4  -4; 4  -1  0; -4 -18  2] [x -1 -4] = [0]

Find the value(s) of x in a matrix equation, involving matrix multiplication, quadratic equations, and complex number solutions. This step-by-step solution includes solving a quadratic equation and interpreting a negative discriminant. Suitable for high school students in Grades 10-12.

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