To find the minimum value of the given expression (3x^2 - 2x - 8), we can use the concept of vertex or completing the square.

Completing the square method:
1. Start with the given quadratic expression: 3x^2 - 2x - 8.
2. Divide the entire expression by the coefficient of x^2 to make the coefficient 1: (3x^2 - 2x - 8)/3.
3. Rearrange the expression by grouping the x terms together: (3x^2 - 2x) - 8/3.
4. To complete the square, take half of the coefficient of x (-2/2 = -1) and square it (1): (3x^2 - 2x + 1) - 1 - 8/3.
5. Simplify the expression: (3x^2 - 2x + 1) - 11/3.
6. Rewrite the expression in factored form: 3(x^2 - (2/3)x + 1/3) - 11/3.
7. Factor the quadratic term inside the parentheses by completing the square: 3(x - 1/3)^2 - 11/3.
8. The minimum value of the expression occurs when the square term is zero, which is at x = 1/3.
9. Substitute x = 1/3 into the factored form of the expression to find the minimum value: 3(1/3 - 1/3)^2 - 11/3 = -11/3.

Therefore, the minimum value of the expression (3x^2 - 2x - 8) is -11/3.

Explanation of the correct answer:
The correct answer is option 'B' (23/3).
It seems there might be an error in the given options, as the correct answer should be -11/3 according to the solution explained above. Please verify the options provided for this question.