Determine which of the following subset of $\mathbb{Q}[x] $ are ideal? This  Question is taken from Hoffman-Kunz books  page No : $138$  2nd edition Pearson publisher 

Determine  which of the following  subset of  $\mathbb{Q}[x]$ are ideal  . When the set is  an ideal , find  its monic generator 

$a)$ all even degree

$b)$ all $f$ such that  $f(2) = f(4) =0$

My attempt :  for option $a)$ If i take $f(x) = x^2 - 4x + 3$ that is even degree ,Then  $f$ will be  monic polynomial that generate the ideal $f \mathbb{Q}[x]$ as such  $f$ is the gcd  for all polynomial  $g$ such that   $g(x) = 0$

Im confused about option $b)$  i was  thinking about $f(x) = (x-2)(x-4)$

Any hints/solution
a) is not an ideal since if $f=x^2+x$ and $g=x^2$ then $f-g=x$ that has odd degree.

b) It is an ideal and it's the set $$\{f(x)\in \mathbb Q[x]\mid f(x)=(x-2)(x-4)g(x)\}=((x-2)(x-4))=...$$
In the first case, $f(x)=x^2$ is a monic generator. 

In the second case, you are right, $f(x)=(x-2)(x-4)$.