We begin by writing the problem as a fraction.

\frac{(2 + 5 i)}{(4 - i)}

Then we multiply the numerator and denominator by the complex conjugate of the denominator.

\frac{(2 + 5 i)}{(4 - i)} \cdot \frac{(4 + i)}{(4 + i)}

To multiply two complex numbers, we expand the product as we would with polynomials (using FOIL).

\begin{matrix} \frac{(2 + 5 i)}{(4 - i)} \cdot \frac{(4 + i)}{(4 + i)} & = & \frac{8 + 2 i + 20 i + 5 i^{2}}{16 + 4 i - 4 i - i^{2}} \\ = & \frac{8 + 2 i + 20 i + 5 (−1)}{16 + 4 i - 4 i - (−1)} & Because i^{2} = −1. \\ = & \frac{3 + 22 i}{17} \\ = & \frac{3}{17} + \frac{22}{17} i & Separate real and imaginary parts . \end{matrix}

Note that this expresses the quotient in standard form.