lg2x+(lg3+lg5)lgx+lg3lg5=0
(lgx)^2 + (lg3+lg5)lgx + lg3lg5 =0
(lgx+lg3)(lgx+lg5)=0
lgx = -lg3 ,x = -1/3
or lgx = -lg5 ,x = -1/5
α*β
= (-1/3)(-1/5)
= 1/15