Let $k$ be a field of characteristic other than $2$. Consider the ring $$A:= k[X,Y]/(X(Y+1),X(Y+X^2)).$$ Describe all the irreducible components of $\operatorname{Spec}(A)$.
It does not look that hard, but I could not solve this on my exam on commutative algebra.
I know that if $\mathfrak{p}\in A$ is a minimal prime ideal, then $V(\mathfrak{p}) = \{\mathfrak{q} \text{ prime ideal } ∣ \mathfrak{p}\subseteq\mathfrak{q} \}$ is an irreducible component of $\operatorname{Spec}(A)$. But how can I use this on a quotient ring?