These are normally really easy for me and this one appears to be incredibly simple but I can't seem to get the correct answer.

Given $\displaystyle x = e^t - 6$ and $\displaystyle y = e^{2t}$

Eliminate the parameter to find a Cartesian equation of the curve.

Isolate t:

$\displaystyle x+6=e^t$

$\displaystyle t=\ln {(x+6)}$

Plug it into the y equation:

$\displaystyle y=e^{2\ln {(x+6)}}$ , $\displaystyle x> {-6}$