Prove that $\mathbb{Q}[X]/(X^4 + X^2)$ is the direct product of two rings.

Question

1. Prove that $R = \mathbb{Q}[X]/(X^4 + X^2)$ is the direct product of two rings.

Proof: I showed that $(X^4+X^2) = (X^2)\cap (X^2+1)$ and $R= (X^2) + (X^2+1)$. According the Chinese Remainder Theorem, we obtain an isomorphism of rings

$$ \mathbb{Q}[X]/(X^4+X^2) \cong \mathbb{Q}[X]/X^2 \times \mathbb{Q}[X]/(X^2+1). $$

2. Compute two non-zero elements $e,f$ of the above ring with the property that $e^2=e$, $f^2=f$, $ef=0$, and $e+f =1$.

3. Find all maximal ideals of $R$.

I'm not sure how to approach 2, or 3. Any help would be appreciated.

I am also wondering if $\mathbb{Q}[X]/(X^2), \mathbb{Q}[X]/(X^2+1)$ are isomorphic to any well-known rings. For example, I know that $\mathbb{Q}[X]/(X) \cong \mathbb{Q}$.