Is there a symbol for plus and minus as opposed to plus or minus?
Another way to write it is:
$$\frac{2}{3a}\left(R_+ + R_-\right) \text{ where } R_\pm = \sqrt[3]{2b^3-9abc\pm\sqrt{-4(b^2-3a)}}$$
There's no single symbol for it. So just use, $$\sum_{n=0}^1\sqrt[3]{2b^3-9abc+(-1)^n\sqrt{-4(b^2-3a)}}$$
No, and there's a good reason for it: it cannot convey the necessary information.
How would the reader know that the intention is to add?
What if you intended the "and" to be for multiplication?
You need to denote the operator somehow, and that will take care of the "and" part by itself.