Answer
$$ x=-4.$$
Work Step by Step
We have
$$
\left(b^{2}\right)^{x+1}=b^{-6} \Longrightarrow b^{2x+2}=b^{-6}.
$$
Since the bases are equal, then the exponents are equal and hence we get
$$2x+2=-6\Longrightarrow 2x=-8 \Longrightarrow x=-4.$$