I am used to identifying and proving periodic signals in the form: x[n] = 3cos(\frac{5n}{7}  +  \frac{ \pi }{4}) with the 'rational/irrational'
method and also getting the fundamental period.

When it comes to this from: x(t) =  e^{(-2+j( \frac{ \pi }{4} ))t} I am not sure how to prove it mathematically. I know that it is a signal with
an envelope function because of the exponential coefficient.

The only step I did for making the signal more clear is: x(t) =  e^{-t}[cos \frac{ \pi t }{4} +jsin \frac{ \pi t}{4}]