The law of the conservation of matter requires that the number of atoms of each element on each side of a chemical equation. This is done by adding coefficients in front of the formulas in the equation. A chemical formula should never be changed, only the amount.

#"AgI+Fe"_2"(CO"_3)_3"##rarr##"FeI"_3+"Ag"_2"CO"_3#

First notice that there is a polyatomic ion, the carbonate ion, #"CO"_3^(2-)"# that occurs on both sides. When there is a polyatomic ion, it is treated as a single entity. So there are three carbonate ions on the left side and one on the right. So a coefficient of #3# is needed in front of #"Ag"_2"CO"_3"#

#"AgI+Fe"_2"(CO"_3)_3"##rarr##"FeI"_3+color(red)("3")"Ag"_2"CO"_3#

Now there are six #"Ag"# atoms on the right side, but one on the left. So a coefficient of #6# needs to be added in front of the compound #"AgI"#.

#color(blue)(6)"AgI+Fe"_2"(CO"_3)_3"##rarr##"FeI"_3+color(red)(3)"Ag"_2"CO"_3#

Now there are six #"AgI"#, which gives us six #"I"# atoms on the left side, but three on the right. So a coefficient of #2# is needed in front of the compound #"FeI"_3"#.

#color(blue)(6)"AgI+Fe"_2"(CO"_3)_3"##rarr##color(green)(2)"FeI"_3+color(red)(3)"Ag"_2"CO"_3#

This also balanced the #"Fe"# atoms. So the balanced equation is

#color(blue)(6)"AgI+Fe"_2"(CO"_3)_3"##rarr##color(green)(2)"FeI"_3+color(red)(3)"Ag"_2"CO"_3#