The only thing I see is perhaps taking it to \(\displaystyle \color{red}\big(\)(a X b) + (a X c)\(\displaystyle \color{red}\big)\) . b but doesn't really help .
You have started off in the right way, but don't forget those parentheses (in red above). When you multiply them out, you get
(a X b) . b + (a X c) . b . But a X b is orthogonal to b, so (a X b) . b = 0, and you are just left with (a X c) . b .