RE Construction: Strings of even length in alphabet {a,b,c} that contain exactly one a
The idea is that if a string contains one $a$, then it can be written as $xay$, where $x,y \in (b+c)^*$. Furthermore, $xay$ has even length if either $x$ has even length and $y$ has odd length, or vice versa. Using this, it is straightforward to come up with a regular expression.