To divide the complex number a_1 + b_1 i by the complex number a_2 + b_2 i, we first set up the fraction $$\frac{a_1 + b_1 i}{a_2 + b_2 i},$$ and then multiply top and bottom by a_2 - b_2 i. This has the effect of making the denominator a real number, and then the division is easy to complete.
\begin{eqnarray} \frac{10 + 5 i}{3 + 4 i}&=&\frac{(10 + 5 i)(3 - 4i)}{(3 + 4 i)(3 - 4i)}\\ &=&\frac{30 - 40 i + 15 i - 20 i^2}{9-16i^2}\\ &=&\frac{50 - 25i}{25}\\ &=&2-i. \end{eqnarray}
The number 3 - 4 i is known as the complex conjugate of 3 + 4 i (and vice versa).