Fill in the coefficients that will balance the following reaction:
(co3)2 + → +

The Coefficients would be:

Hg(CO3)2: 1
AgCl: 4
HgCl4: 1
Ag2CO3: 2

How do we know this?
Because if we look at the coefficent for the first compound, Hg(CO3)2, That means in total we have one Hg and 2 CO3's. 

IF the coefficient is 4 for the next compound, AgCl, there are 4 Ag's and 4 Cl's.

Those were the REACTANTS to our equation. Now looking at the arrow, HgCl4 and Ag2CO3 are our PRODUCTS, or, what is FORMED from Hg(CO3)2 and AgCl. So, we need to make sure that they are equal, to obey the law of conservation of mass. So in simpler terms, if we have 4 Cl's on the reactant side, we need 4 on the products side.

So, if we had the coefficient of 1 for HgCl4, this makes sense, because on the products side we had 1 Hg, and 4 Cl's. With the coefficient of 1, we have 1 Hg and 4 Cl's on the products side, as well!

And if we had the coefficient of 2 for Ag2CO3, we would have 4 Ag's total and 2 CO3s, which matches on the reactants side, for we have two CO3's and 4 Ag's.

I hope I could help!