Answer
$x=-\frac{2}{3}$
Work Step by Step
We rewrite the right side, so that the base is $\frac{1}{e}$ on both sides.
$\frac{1}{e}^{-x}=[(\frac{1}{e})^2]^{x+1}$
By the law of exponents: $(a^x)^y=a^{xy}$
$((\frac{1}{e})^2)^{x+1}=(\frac{1}{e})^{2x+2}$
The rewritten equation:
$\frac{1}{e}^{-x}=(\frac{1}{e})^{2x+2}$
Then, since the bases are the same, we set the exponents equal, because the two sides are equal only if the powers are equal too: if $b^m=b^n$ and $b \ne0$ , $b \ne1$ then $m=n$
$-x=2x+2
\\-x-2x=2$
$-3x=2
\\\frac{-3x}{3}=\frac{2}{-3}$
$x=-\frac{2}{3}$