\displaystyle \begin{align}(1+2i)\left(2-\frac{i}{4}\right)&=1\cdot2-1\cdot\frac{i}{4}+2i\cdot2-2i\cdot\frac{i}{4}=\\&= 2-\frac{i}{4}+4i+\frac{1}{2}=\\&=\frac{5}{2}+\frac{15i}{4}\end{align}
