To find the x-intercepts of the quadratic equation \( y = x^2 + 3x + 8 \), we need to set \( y = 0 \) and solve for \( x \).
The equation becomes:
\[ 0 = x^2 + 3x + 8 \]
To determine the x-intercepts, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, \( a = 1 \), \( b = 3 \), and \( c = 8 \).
Calculating 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 solutions for \( x \), indicating that there are no x-intercepts.
Thus, the correct response is:
There is no x-intercept.