write an equation that has a root of 3+i.

If it is to have real coefficients, then if one of its
roots is 3+1, it must have 3-i as another root.  We can 
make an equation with just those two roots:

If we were solving it and ended up with

x = 3 + i   and x = 3 - i

Then to get that we would have had to set two expressions = 0
using the zero factor principle. So before that we would have
had:

x - 3 - i = 0  and x - 3 + i = 0

Then before that we must have had

(x - 3 - i)(x - 3 + i) = 0

And before that we must have had

x² - 3x + ix - 3x + 9 - 3i - ix + 3i - i² = 0

Collecting terms:

x² - 6x + 9 - i² = 0

Replacing i² by -1

x² - 6x + 9 - (-1) = 0

x² - 6x + 9 + 1 = 0

x² - 6x + 10 = 0 

That's it.

Edwin