Weird. I didn't look through the whole thing.
Compare forms for the exponential:
General:
$\displaystyle e^{j( \omega _0 t + \theta )}$
Yours:
$\displaystyle e^{-(2 + 3j)t}$
Now we can rewrite yours as $\displaystyle \sqrt{26} e^{j(-3t)} \cdot e^{-2t} = \left ( \sqrt{26} e^{-2t} \right ) e^{-3jt}$
It would appear that your signal has an extra time varying amplitude which has the effect of a decay term. (That or you could look at it as having a complex phase term.)
-Dan
this has more than the answer to your question but as an (I assume) an up and coming electronics engineer you need to know all of this.
https://www.electronics-tutorials.ws...x-numbers.html
I've looked over that tutorial and it was very helpful. I guess the very last thing that I am confused about is the e^-2t term. What is it and why is it not a part of the answer for the amplitude?
It's what's known as a decay term. It is part of the amplitude actually. In this case the amplitude is time varying.
If you have a sinusoidal signal $s(t) = |A|e^{j 2\pi f t + \phi}$
And a purely real function $e(t)$ then
$e(t)s(t) = |A e(t)|e^{j 2\pi f t + \phi}$
i.e. the real function contributes only to the amplitude, not to the phase.