How can I balance the following equation atomically and electrically?
Deal with the two half equations separately and then combine them.

Starting with the oxalate equation: $$\ce{C2O4^{2-} -&gt; CO2}$$

Balance the atoms: $$\ce{C2O4^{2-} -&gt; 2CO2}$$

Now balance the charge by adding electrons: $$\ce{C2O4^{2-} -&gt; CO2 + 2e-}$$

Now for the manganese dioxide reduction: $$\ce{MnO2 -&gt; Mn^{2+}}$$

We can balance the half equations by adding water, hydrogen ions or electrons. Since water is the only one which contains oxygen we should add this first: $$\ce{MnO2 -&gt; Mn^{2+} + 2H2O}$$

Now add hydrogen ions to balance the hydrogens: $$\ce{MnO2 +4H+ -&gt; Mn^{2+} + 2H2O}$$

Finally add electrons the balance the charge: $$\ce{MnO2 +4H+ + 2e- -&gt; Mn^{2+} + 2H2O}$$

Now combine the equations to cancel all the electrons. In this case there are two on both sides so we don't need to multiply any of the equations by anything.

$$\ce{MnO2 +4H+ + C2O4^{2-} -&gt; Mn^{2+} + 2CO2 + 2H2O}$$