Finding an unknown coefficient of a polynomial given a factor Q:Find the value of $a$ given that $x^2+1$ is a factor of $x^4-3x^3+3x^2+ax+2$
No idea where to start, I was going to use the factor theorem but it didn't work out. 
Question from year 10 Cambridge maths textbook

Divide your polynomial by $x^{2} + 1$ and set the remainder equal to zero.
Or, since you are only interested in the remainder, set $x^{2} = -1$ in the polynomial to get that the remainder is
$$
(x^2)^{2}-3 x x^2+3 x^2+ax+2
=
(-1)^{2} + 3 x - 3 + a x + 2
=
(a + 3) x.
$$
So $x^{2} + 1$ divides your polynomial iff $a = -3$.
