I checked with WolframAlpha, and it states the same thing as I typed in (implying it cannot be factored). GeoGebra and Symbolab are saying similar results.
However, there might be some clever trick that I'm not thinking of, so I'll let another tutor offer a second opinion.
You can put this solution on YOUR website! Factorize 2a⁴b^6-3a²b²-20
=============
Not possible, IMO.
If you meant
-------------
You can sub x for a^2*b^3 and get
2x^2 - 3x - 20
======================
(2x + 5)*(x - 4)