The book I'm reading defines an ideal generated by $\langle a_1, ..., a_n \rangle$ as $I = \langle a_1, ..., a_n \rangle = \{r_1a_2 + ... + r_na_n: r_i \in R\}$.
So for $Z[x]$, the ring of all polynomials with integer coefficients, and $I$ (the subset of $Z[x]$ with even constant terms), $I$ is an ideal and $I = \langle x, 2 \rangle$.
But if $r_1 = 3x^2 + 2x$ and $r_2 = x^3 + 1$ , then $(3x^2+2x)x + (x^3+1)2 = 5x^3 + 2x + 2$ which is not in $I$.
What am I misunderstanding here?
Your ideal $I$ is just the set of polynomials of the form $$f(X)=XP(X)+2Q(X) \qquad P,Q \in \Bbb Z[X]$$ It is equal to $$\{2a_0+a_1X+\cdots+a_nX^n \mid a_i \in \Bbb Z, n ≥ 0\}$$
Your polynomial $5X^3+2X+2$ has this form, with $a_3=5,a_1=2,a_2=0,a_1=1$. So it belongs to $I$.