To find the conjugate pair solutions to the equation \( x^2 - 3x + 8 = -2 \), we first rewrite the equation in standard form:
\[ x^2 - 3x + 8 + 2 = 0 \implies x^2 - 3x + 10 = 0. \]
Next, we can apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]
where \( a = 1 \), \( b = -3 \), and \( c = 10 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4(1)(10) = 9 - 40 = -31. \]
Since the discriminant is negative, we can conclude that there are two complex conjugate solutions. We now can substitute back into the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{-31}}{2(1)} = \frac{3 \pm \sqrt{-31}}{2} = \frac{3}{2} \pm \frac{i\sqrt{31}}{2}. \]
Thus, the solutions can be written as:
\[ \frac{3}{2} \pm \frac{i\sqrt{31}}{2}. \]
So the correct answer showing the conjugate pair solutions for the equation \( x^2 - 3x + 8 = -2 \) from the provided options is:
Negative \(\frac{3}{2}\) plus or minus \(\frac{i\sqrt{31}}{2}\).