Questions in algebraic geometry related to finding $V_A(X)$

This question is from Algebraic Geometry assignment and I was unable to solve some of these:

(a) I have solved.

(b) Prove that the polynomial $X^3 +X^2 +X+1 \in \mathbb{Z}/ \mathbb{Z}4[X]$ is a multiple of $X+1$ and $X+3$ but not of $(X+1) (X+3)$.

I proved that it is a multiple of $X+1$ and $X+3$ by showing that $X^3+X^2 +X+1$ has both $-1$ and $-3$ as a root but I am unable to understand why it cannot have both as root simultaneously.

(c) Give an example of a commutative ring $A$ such that $V_A (X^2 -X) $ is infinite.

I am unable to find such a ring. There must be infinite number of elements satisfying $X^2=X$. So, ring must be infinite. examples are : $\mathbb{Z}$ , $\mathbb{Q}$ but they don't satisfy the given condition. So, please help.

(d) and (e) I have done.

Kindly help.

For the first two questions, simply note that \begin{align*} P(X)=X^3+X^2+X+1 & = (X+1)(X^2+1) \\ & = (X+3)(X^2 + 2X + 3) \\ & = (X+1)(X+3)\cdot (X+1) + 2X+2\end{align*}

The remainder is zero in the first two cases, not in the third one.

For your question (c), a ring like $(\mathbb{Z}/2\mathbb{Z})^{\mathbb{N}}$ fits the bill: every single element $x$ inside it is such that $x^2=x$ (and it is infinite).