There are several conditions for something to be a space. For instance,
if A and B are in the space, is A+B in the space. Also, if (e.g.) A is in the space, is 3A in the space?
To start off, for your case, if p(t) is at^2, and q(t) is bt^2, is p(t)+q(t) = ct^2 for some c?