We can express the factors as (5x√5+a)(x+b) where ab=8√5, so b=8√5/a, and 5b√5+a=30, the coefficient of the middle term. Therefore (5√5)(8√5/a)+a=30, that is, 200/a+a=30; 200+a^2=30a and a^2-30a+200=0=(a-20)(a-10).
Therefore, b=8√5/20=2√5/5 or b=4√5/5 and the factors are:
(5x√5+20)(x+2√5/5) or (5x√5+10)(x+4√5/5).
These factorisations are equivalent: the middle term, 30x, is split into (5√5)(2√5/5)x+20x=10x+20x, or (5√5)(4√5/5)x+10x=20x+10x.