complex long division

Question

For example we have $(2+7i)(4-i)=15+26i$.

What I am after is some kind of long division method so that:

$(2+7i)|\overline{15+26i}=x+yi$

If we guess $x=4$ we get a remainder of $7-2i$, but is there (or not) a structured method to solve complex long division problems?

EDIT:

I think we can use simultaneous equations:

$$2x+7y=15$$ $$7x-2y=26$$

Answer 1

You are better off doing this $$ z_1z_2 = z_3 $$ then we have $$ z_1 = \frac{z_3}{z_2} = \frac{z_3\cdot z_2^*}{z_2\cdot z_2^*} $$ where ($^*$) is the complex conjugate. so the denominator is just $$ z_2\cdot z_2^* = |z_2|^2 $$ taking your example $$ \frac{15+26i}{2+7i} = \frac{(15+26i)(2-7i)}{(2+7i)(2-7i)} = \frac{212-53i}{53} = 4-i $$