In the ring $A = k[X]/(X^2(X+1)^3)$, why ideal $(X)^2 \subset (X)^3$ holds?

Question

$(X)^2, (X)^3$ are ideal products and $ (X)^3 \subseteq (X)^2$ holds in any ring.

Answer 1

In $A$, $$X^2=0+X^2=(-X^2(X+1)^3)+X^2=-X^3(X^2+3X+3)$$ Therefore $(X^2)=(X^3)$.