Which answer shows the conjugate pair solutions to the equation x^2 - 3x + 8 = -2?

To find the conjugate pair solutions to the equation \( x^2 - 3x + 8 = -2 \), we first simplify the equation:

\[ x^2 - 3x + 8 + 2 = 0 \]

This simplifies to:

\[ x^2 - 3x + 10 = 0 \]

We can use the quadratic formula to solve for \( x \), which is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = -3 \), and \( c = 10 \):

1. Calculate \( b^2 - 4ac \):

\[ b^2 = (-3)^2 = 9 \] \[ 4ac = 4 \cdot 1 \cdot 10 = 40 \] \[ b^2 - 4ac = 9 - 40 = -31 \]

2. Substitute into the quadratic formula:

\[ x = \frac{3 \pm \sqrt{-31}}{2} \]

3. Simplifying gives:

\[ x = \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \]

Thus, the conjugate pair solutions are:

\[ \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \]

Looking at the provided options, the correct answer is:

D. \(\frac{3}{2} \pm \frac{i\sqrt{31}}{2}\)