they are linearly dependent ,is equavalent to say:
Two vectors are "dependent" if one is a multiple of the other (more generally, a set of vectors is dependent with there exist a linear combination of the vectors, with not all coefficients equal to 0, that is equal to 0. For two vectors, u and v, this says that au+ bv= 0 and then either u= (b/a)v, if a is not 0, or v= (a/b)v if it is.
If your vectors are <2, 2t+6> and <-t, 4t>, then they are dependent if and only if <2, 2t+6>= a<-5, 4t> for some a. That is saying that 2= -at and that 2t+ 6= 4at. From the first equation, a= -2/t so the second equation becomes 2t+ 6= 4(-2/t)t= -8. Solve that for t.
If your vectors are <2, -t> and <2t+6, 4t>, then they are dependent if and only if <2, -t>= a<2t+6, 4t> for some t. That is saying that 2= 2at+ 6a and -t= 4at. From the second equation, 4a= -1 so a= -1/4 and the first equation becomes 2= -(1/2)t- 3/2. Solve that for t.