Let <a1,a2,a3,a4> be an element in thissubset.

Does it mean that k<a1,a2,a3,a4>=<ka1,ka2,ka3,ka4> is in thissubset?

Yes, because,

a1*a2*a3*a4 >= 0

Means that,

(ka1)*(ka2)*(ka3)*(ka4)=k^4(a1*a2*a3*a4)>= 0 because k^4>=0.

And if <b1,b2,b3,b4> is another element insubset.

Does it mean that <a1+b1,a2+b2,a3+b3,a4+b4> in thesubset?

No, because,

Consider,

<1,2,3,4> and <-2,-1,3,4>

Yet, their sum,

<-1,1,6,8> is not an element because (-1)(1)(6)(8)<0.

Thus, this does not form asubspace.