To find the roots of the quadratic equation \( g(x) = 4x^2 + 256 \), we need to set \( g(x) = 0 \):
\[ 4x^2 + 256 = 0 \]
Subtracting 256 from both sides, we have:
\[ 4x^2 = -256 \]
Next, divide by 4:
\[ x^2 = -64 \]
To solve for \( x \), we take the square root of both sides:
\[ x = \pm \sqrt{-64} \]
Since the square root of a negative number involves the imaginary unit \( i \):
\[ x = \pm 8i \]
Thus, the roots of the equation \( g(x) = 4x^2 + 256 \) are:
\[ -8i \quad \text{and} \quad 8i \]
When writing the roots in order from smaller to larger, we get:
\[ -8i, 8i \]
So, the exact answers are:
\[ -8i, 8i \]