Well if a quadratic has roots
α and β then it can be written as
(x−α)(x−β).
Think about it, when you solve a quadratic, you set it to 0 and then you factorise and find the values of x for when each of the factors is 0. So you know that a quadratic with roots 2 and 3 can be written as (x-2)(x-3) because if you had (x-2)(x-3)=0 you would have no problem telling me the roots.
From this it follows that if one the roots,
α, is
a+bi then the quadratic can be written as
(x−(a+bi))(x−β).
If a quadratic (or any degree polynomial) with
real coefficients has complex root
w=a+bi then its complex conjugate
w∗=a−bi is also a root of the quadratic (this holds true for any degree polynomial with all real coefficients).
From this we know that a-bi is also a root of the quadratic equation so we can say that
β=a−bi so the quadratic can be written as
(x−(a+bi))(x−(a−bi)).