Answer
$x=-\displaystyle \frac{1}{2}$
Work Step by Step
First, we restrict the solutions to $x+1\gt 0$, that is, to
$x\gt -1,\qquad(*)$
otherwise the equation would not be defined, as logarithmic functions are defined only for positive arguments.
We can use the definition of logarithm here:
$\log_{2}(x+1)=-1$ means $2^{-1}=x+1$
$x+1=\displaystyle \frac{1}{2}$
$x=-\displaystyle \frac{1}{2}$, which satisfies the condition (*), so it is a valid solution.
