Let Z be the set of integers.

$\mathrm{A}=\left\{\mathrm{x} \in \mathrm{z}: 2^{(\mathrm{x}+2)\left(\mathrm{x}^{2}-5 \mathrm{x}+6\right)}=1\right\}$

$2^{(x+2)\left(x^{2}-5 x+6\right)}=2^{0} \Rightarrow x=-2,2,3$

$A=\{-2,2,3\}$

$B=\{x \in Z:-3<2 x-1<9\}$

$B=\{0,1,2,3,4\}$

$A \times B$ has is 15 elements so number of subsets of $A \times B$ is $2^{15}$.