Regular expression for language with even number of 0's and 1's
Constructing regular expressions is usually more complicated than programming finite state automata. The standard way would be to construct a FSA and translate that one into a regular expression using standard techniques.

But if you prefer, it can be done directly, in two steps. Start by making two regular expressions $\alpha$ and $\beta$. 


First $\alpha$ specifies strings with even number of $1$'s, and exactly two
$0$'s, one of which is the last symbol of the string. 
Similarly $\beta$ specifies strings with odd number of $1$'s, and exactly two $0$'s,
one of which is the last symbol of the string.


Then build an expression for strings that have an even number of $0$ ánd an even number of $1$'s based on  $\alpha$ and $\beta$.