To find the x-intercepts of the quadratic equation \( y = x^2 + 3x + 8 \), we set \( y \) to 0 and solve for \( x \):
\[ 0 = x^2 + 3x + 8 \]
Next, we can apply the quadratic formula, which is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \( x^2 + 3x + 8 \), we have \( a = 1 \), \( b = 3 \), and \( c = 8 \).
Now, let's calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 8 = 9 - 32 = -23 \]
Since the discriminant is negative, the quadratic equation has no real roots. This means there are no x-intercepts.
Thus, the correct answer is:
There is no x-intercept.