a, b and c are unknowns and I have two equations: $a-b=x_1$ and $b+c=x_2$. Is there a way to obtain explicit solution to $ab$ or $abc$ or any other multiplication of the unknowns?
Any approximations are welcome. I am not looking for an exact solution.
Edit: please note that I am not looking for solutions for $a$, $b$ or $c$, but for $ab$ or $bc$ or any other combination.
It's impossible, because $ab$ is not uniquely determined by this system.
The general solution of the system is $$a = t, \quad b = t - x_1, \quad c = x_1 + x_2 - t$$ where $t$ is any real number. Then $ab = t^2 - tx_1$, and you can see that by varying $t$, this may take different values.
Explicitly, suppose $x_1 = x_2 = 0$, so the system is $a-b=0$ and $b+c=0$. Then one solution is $a=2, b=2, c=-2$, in which case $ab=4$. Another solution is $a=3, b=3, c=-3$ in which case $ab = 9$. So the system of equations does not give you enough information to tell whether $ab$ is $4$ or $9$ or some other value.
One thing you can say: by completing the square, we have $$ab = \left(t - \frac{1}{2}x_1\right)^2 - \frac{1}{4} x_1^2 \ge -\frac{1}{4} x_1^2$$ so at least you have a lower bound for $ab$.