Question

e$e^{x} - e^{-2} =-2$

Answer 1

$e\cdot e^x -e^{-2}=-2$

$\implies e^{x+1}=e^{-2}-2$

note that RHS is negative. (because e with negative exponent is less than 1)

and LHS is always positive.

so there cannot be any solution