To find the x-intercepts of the quadratic function \( y = x^2 + 3x + 8 \), we need to set \( y = 0 \) and solve for \( x \):
\[ 0 = x^2 + 3x + 8 \]
To solve this quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1 \), \( b = 3 \), and \( c = 8 \). First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 3^2 - 4(1)(8) = 9 - 32 = -23 \]
Since the discriminant is negative (\(-23\)), this means there are no real x-intercepts.
Therefore, the correct answer is:
There is no x-intercept.