Let's start from the final situation. After the two spheres are connected with the conducting wire, the total charge distributes equally between the two spheres (because they are identical). We can call the charge on each sphere 
, with Q being the total charge.
The electrostatic force in this situation is 0.0360 N, so we can write
where k is the Coulomb's constant and
is the separation between the two spheres. Using F=0.0360 N, we can find the value of Q, the total charge shared between the two spheres:
Now let's go back to the initial situation, before the conducting wire was attached; in this situation, the two spheres have a charge of
and 
, whose sum is Q:
The electrostatic force between the two spheres in the initial situation is:
And since we know F=0.108 N, we find
But the problem tells us that the two spheres have charges of opposite sign, so we must put a negative sign:
So now we have basically a system of 2 equations:
If we solve it, we find the initial charge on the two spheres:
The electrostatic force in this situation is 0.0360 N, so we can write
where k is the Coulomb's constant and
Now let's go back to the initial situation, before the conducting wire was attached; in this situation, the two spheres have a charge of
The electrostatic force between the two spheres in the initial situation is:
And since we know F=0.108 N, we find
But the problem tells us that the two spheres have charges of opposite sign, so we must put a negative sign:
So now we have basically a system of 2 equations:
If we solve it, we find the initial charge on the two spheres: