Comment
I'm afraid your definitions of a memoryless/memory system are incorrect. If a system is memoryless, that simply means that the output depends only on the input at the current moment in time (and not on any past inputs). Therefore, the two examples you used are actually both memoryless. The $ t $ terms are not considered to cause the system to have memory because at each $ t $, $ t $ is simply a constant and not part of the input $ x(t) $. - Jacob Pfister
I'm not really sure if $ x(t)^2\, $ means you square the whole $ x(t)\, $, or you square the value of $ t\, $ only. If its the first case, then its a memoryless system, but if its the latter, then i'm afraid its a system with memory.
Also, $ (t-2)^2\, $ doesn't depend on the future. $ t\, $ is just a certain variable representing time, thus we can't prove its a system with memory if $ (t-2)^2\, $ appears. This phrase, $ (t-2)^2\, $ is a memoryless system - Wei Jian Chan
