To find the roots of the function \( f(x) = 6x^2 + 216 \), we can set \( f(x) \) equal to zero and solve for \( x \):
\[ 6x^2 + 216 = 0 \]
Subtract 216 from both sides:
\[ 6x^2 = -216 \]
Now, divide both sides by 6:
\[ x^2 = -36 \]
To solve for \( x \), take the square root of both sides:
\[ x = \pm \sqrt{-36} \]
Since \(\sqrt{-36}\) involves the imaginary unit \( i \):
\[ x = \pm 6i \]
Thus, the roots of the function are:
\[ \pm 6i \]
So, the correct response is \( \pm 6i \).