complex long division 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:
$$2x+7y=15$$
$$7x-2y=26$$

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
$$
\frac{15+26i}{2+7i} = \frac{(15+26i)(2-7i)}{(2+7i)(2-7i)} = \frac{212-53i}{53} = 4-i
$$
