= Solution
Applying the <Itô formula> to $f(x)=x^2$ and the semimartingale $B$ gives
$$
B_t^2=2\int_0^tB_s\,dB_s+t.
$$
Hence
$$
X_t=-2\int_0^tB_s\,dB_s+\int_0^t(B_s^2-1)\,ds.
$$
The first term is a continuous local martingale and the second has finite variation. By uniqueness of the continuous semimartingale decomposition, $X$ could be a local martingale only if the finite-variation term were constant. Its derivative $B_s^2-1$ is not zero almost everywhere, so $X$ is not a local martingale.
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