To find the roots of the polynomial function \( h(x) = 3x^2 + 243 \), we can set the equation equal to zero and solve for \( x \):
\[ 3x^2 + 243 = 0 \]
Subtracting 243 from both sides gives:
\[ 3x^2 = -243 \]
Dividing both sides by 3 yields:
\[ x^2 = -81 \]
Taking the square root of both sides, we find:
\[ x = \pm \sqrt{-81} \]
The square root of \(-81\) can be expressed in terms of \(i\) (the imaginary unit):
\[ x = \pm 9i \]
Thus, the roots of the polynomial function are \( 9i \) and \( -9i \).
The correct response is:
The roots are 9i and −9i.