There seems to be one character missing. But I gather that <em>f(x)</em> needs to satisfy
• divides
• divides
I'll also assume <em>f(x)</em> is monic, meaning the coefficient of the leading term is 1, or
Since divides
, and
where is degree-2, and we can write it as
Now, we have
so if divides
, then
is degree-2, so
is degree-4, and we can write
where is degree-1.
Expanding the left side gives
and dividing by leaves no remainder. If we actually compute the quotient, we wind up with
If the remainder is supposed to be zero, then
Adding these equations together and grouping terms, we get
Then , and you can solve for <em>a</em> and <em>b</em> by substituting this into any of the three equations above. For instance,
So, we end up with