This quest was marked obsolete by Blizzard and cannot be obtained or completed.
You take a sample of slime from Un'Goro Crater and place some of it into the various tubes and beakers on the table. As things begin to boil and spurt, the slime changes colors. When the process slows down and the liquids calm, you see a small petri dish with your purified Un'Goro slime samples in it along with some other remnants.
You pick up the petri dish with the slime sample in it. The only way to discern if the samples is pure enough is to open it and examine the contents yourself. Ideally you'll find a sample that fits Chemist Fuely's criteria.