To find the roots of the function \( g(x) = 4x^2 + 256 \), we can set the equation equal to zero:
\[ 4x^2 + 256 = 0 \]
Next, we isolate \( x^2 \) by subtracting 256 from both sides:
\[ 4x^2 = -256 \]
Dividing both sides by 4 gives:
\[ x^2 = -64 \]
Taking the square root of both sides, we remember to include both the positive and negative roots:
\[ x = \pm \sqrt{-64} \]
Since \( \sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i \) (where \( i \) is the imaginary unit), the roots can be expressed as:
\[ x = 8i \quad \text{and} \quad x = -8i \]
Listing the roots, we have:
\[ -8i \quad \text{and} \quad 8i \]
Therefore, the exact roots of the function are:
\[ \boxed{-8i} \quad \text{and} \quad \boxed{8i} \]
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