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To find the solutions to the equation \(3x^2 - 4x + 6 = 2\), we start by simplifying and solving the quadratic equation.

1. **Simplify the equation**:
   \[
   3x^2 - 4x + 6 = 2
   \]
   Subtract 2 from both sides:
   \[
   3x^2 - 4x + 4 = 0
   \]

2. **Use the quadratic formula**:
   The quadratic formula is:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   For our equation \(3x^2 - 4x + 4 = 0\), \(a = 3\), \(b = -4\), and \(c = 4\).

3. **Calculate the discriminant**:
   \[
   b^2 - 4ac = (-4)^2 - 4 \cdot 3 \cdot 4 = 16 - 48 = -32
   \]
   The discriminant is \(-32\), which is negative, indicating that the solutions will be complex numbers.

4. **Find the solutions**:
   \[
   x = \frac{-(-4) \pm \sqrt{-32}}{2 \cdot 3} = \frac{4 \pm \sqrt{-32}}{6}
   \]
   Simplify \(\sqrt{-32}\):
   \[
   \sqrt{-32} = \sqrt{-1 \cdot 32} = \sqrt{32} \cdot i = 4\sqrt{2}i
   \]
   Therefore, the equation becomes:
   \[
   x = \frac{4 \pm 4\sqrt{2}i}{6} = \frac{4}{6} \pm \frac{4\sqrt{2}i}{6} = \frac{2}{3} \pm \frac{2\sqrt{2}i}{3}
   \]
   Simplifying further:
   \[
   x = \frac{2 \pm 2\sqrt{2}i}{3}
   \]

The correct solution from the given options is:
\[
x = \frac{2 \pm \sqrt{8}i}{3} = \frac{2 \pm 2\sqrt{2}i}{3}
\]
So, the correct answer is:
\[
\boxed{x = \frac{2 \pm \sqrt{8}i}{3}}
\]

Would you like further details on any specific step or have any other questions? Here are some related questions you might find interesting:

1. How do you determine if a quadratic equation has real or complex solutions?
2. What are the steps to solve a quadratic equation using the quadratic formula?
3. How do you simplify the square root of a negative number?
4. What is the significance of the discriminant in the quadratic formula?
5. Can you explain why complex solutions occur when the discriminant is negative?
6. How do you interpret the plus-minus (\(\pm\)) symbol in the quadratic formula?
7. How would you solve a quadratic equation by completing the square?
8. What are some practical applications of solving quadratic equations with complex solutions?

**Tip:** Always double-check the discriminant calculation, as it determines the nature of the roots (real or complex).

Learn how to solve the quadratic equation 3x^2 - 4x + 6 = 2 with a detailed step-by-step solution using the quadratic formula and understanding complex numbers. This problem covers key algebraic concepts and the discriminant theorem. Suitable for high school students (Grades 10-12).

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