The trick is to use the complex conjugate of the denominator to make the denominator real. And then simplify.

\(\displaystyle{\frac{{{1}-{i}}}{{{2}+{4}{i}}}}={\frac{{{1}-{i}}}{{{2}+{4}{i}}}}\times{\frac{{{2}-{4}{i}}}{{{2}-{4}{i}}}}={\frac{{{2}-{4}{i}-{2}{i}-{4}}}{{{4}+{16}}}}={\frac{{-{2}-{6}{i}}}{{{20}}}}={\frac{{-{1}-{3}{i}}}{{{10}}}}\)

\(\displaystyle{\frac{{{1}-{i}}}{{{2}+{4}{i}}}}={\frac{{{1}-{i}}}{{{2}+{4}{i}}}}\times{\frac{{{2}-{4}{i}}}{{{2}-{4}{i}}}}={\frac{{{2}-{4}{i}-{2}{i}-{4}}}{{{4}+{16}}}}={\frac{{-{2}-{6}{i}}}{{{20}}}}={\frac{{-{1}-{3}{i}}}{{{10}}}}\)