First, we start with the given equation:
\[ -2x^2 - 8x = -8x + 8 \]
Next, we can simplify the equation by moving all terms to one side. We can add \(8x\) to both sides:
\[ -2x^2 - 8x + 8x = 8 \]
This simplifies to:
\[ -2x^2 = 8 \]
Now, we can divide both sides by \(-2\):
\[ x^2 = -4 \]
Next, we take the square root of both sides. Remember, the square root of a negative number involves the imaginary unit \(i\):
\[ x = \pm \sqrt{-4} = \pm 2i \]
Thus, the solutions to the equation are:
\[ x = 2i \quad \text{and} \quad x = -2i \]
The final answer expressed as complex numbers with the imaginary unit is:
\[ x = 2i \quad \text{or} \quad x = -2i \]
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