Problem 1: Show that the ideal $$I:=(af+d^2,bf+de,cf+e^2)+(a,b,c)^2+(a,b,c)(d,e)$$ is primary in the polynomial ring $k[a,b,c,d,e,f]$ where $k$ is a field.
Macaulay2 confirmed the answer. But I struggle to show it on paper. It is clear that the radical of this ideal is $(a,b,c,d,e)$ which is prime. Using the Grobner basis obtained by Macaulay2 and the results in https://www.sciencedirect.com/science/article/pii/S0747717188800403. It suffices to prove the following.
Problem 2: Show that if we have two polynomials $g,h\in k[a,b,c,d,e,f]$ such that $g=f^nh$ for some integer $n$ and $g\in I$, then $h\in I$.
I feel like Problem 2 can be solved with Grobner bases, which are obtainable with computers, but my lack of experience with this tool prevents me from proceeding.
On the other hand, I still hold out some hopes that one can solve Problem 1 directly, since it has a nice structure.
Thank you!!!