To divide \(-510\) by \(34\), we perform the calculation:
\[ -\frac{510}{34} = -15 \]
However, since \(510\) and \(34\) do not simplify to any fractional part and safely fall within the integer scope, none of the given options \(23, 2 \text{ thirds}, -23, \text{negative } 2 \text{ thirds}, -38,\) or \text{negative } 3/8 are correct.
Thus, the correct option isn't in the provided responses. The correct answer is \(-15\).