To solve the equation \( x^2 - 3x + 8 = -2 \), we start by rewriting it in standard form:
\[ x^2 - 3x + 10 = 0 \]
Next, we can use the quadratic formula to find the solutions, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 1 \), \( b = -3 \), and \( c = 10 \). Now we compute the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4(1)(10) = 9 - 40 = -31 \]
Since the discriminant is negative (\(-31\)), this means the solutions will be complex.
Now we substitute the values into the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{-31}}{2(1)} = \frac{3 \pm i\sqrt{31}}{2} \]
This can be expressed as:
\[ x = \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \]
Thus, the conjugate pair solutions to the equation are:
\[ \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \]
From the provided responses, the correct answer is:
\[ \text{Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 31 End Root over 2 End Fraction} \]