What is a "symmetrical product" of an irreducible representation with itself?
In your case you need the equations for the characters as there are symmetric and antisymmetric parts in the product. Your question gives just the symmetric part and the antisymmetric part is $A_2$ normally written as $[A_2]$ to indicate this. 

The characters are found using $\chi^+ =\left([\chi(R)]^2+ \chi(R^2) \right)/2$ for the symmetrical part and $\chi^- =\left([\chi(R)]^2- \chi(R^2) \right)/2$ for the antisymmetric part. Using the point group table you need to work out the square of all the operations for $E_g$ this gives $\chi(R^2)$ (e.g. $(C_2)^2 \to E$) and then square the operations which gives $[\chi(R)]^2$, i.e. $[E]^2\to 4$. From the list of characters you then identify the symmetry species, either directly by inspection (as for the antisymmetric in this case), or by forming the irreps from the reducible representation.