1. Find set $A$

Given $A = \{x \in \mathbb{Z} : 2(x+2)(x^2-5x+6)=1\}.$

Since $x \in \mathbb{Z}$, the left-hand side $2(x+2)(x^2-5x+6)$ is an even integer, because it has a factor $2$.

But the equation says $2(x+2)(x^2-5x+6)=1,$ and $1$ is odd. An even integer can never equal $1$.

Hence, there is no integer solution.

So, $A=\varnothing.$

2. Find set $B$

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

Solve the double inequality: $-3 < 2x-1 < 9.$

Add $1$ throughout: $-2 < 2x < 10.$

Divide by $2$: $-1 < x < 5.$

Since $x \in \mathbb{Z}$, $x=0,1,2,3,4.$

Therefore, $B=\{0,1,2,3,4\}, \quad |B|=5.$

3. Find $A \times B$

We know that $A=\varnothing.$

Therefore, $A \times B = \varnothing \times B = \varnothing.$

So, $|A \times B|=0.$

4. Number of subsets of $A \times B$

A set with $n$ elements has $2^n$ subsets.

Here $n=0$, so number of subsets is $2^0=1.$

5. Compare with options

The derived answer is $1,$ which does not match any of the given options $2^{12}, 2^{18}, 2^{10}, 2^{15}$ (written in the prompt as 212, 218, 210, 215).

This suggests there is likely a typo in the question statement for set $A$.

If the intended condition for $A$ were something else, then the stored answer $2^{15}$ would correspond to $|A\times B|=15.$ Since $|B|=5$, this would require $|A|=3$. But with the given equation $2(x+2)(x^2-5x+6)=1$, we definitely get $A=\varnothing$.

So the stored answer cannot be correct for the question as written.