Price equilibria with transfers

I am pretty new to Microeconomics so please bear with me if I am missing something obvious.
I am solving the following problem:

The question is in the setting of pure exchange economy with two consumers A, and B who have the utility functions:

$u_A(x_{1A}, x_{2A}) = x_{1A} x_{2A}$,
$u_B(x_{1B}, x_{2B}) = x_{1B} + 2x_{2B}$.
It is assumed that the endowments of both customers is positive for both commodities.

(a) Sketch the Edgeworth box, indifference curves of the consumers and the Pareto set where $bar{omega_1} = omega_{1A} + omega_{2A} = omega_{1B} + omega_{2B} = bar{omega_2}$.

I have solved this part and the Edgeworth box looks like this:

Next it is asked (b) wether all the Pareto Optima of this economy can be sustained as price quasi-equilibria with transfers. I know that I should us the second welfare theorem but I am not sure how I can check the convexity of the preferences using utilities.