I'm confused with some ideas about quotient rings and wanted to check if my intuition was flawed in any ways:
1: $$\mathbb{Z}[\sqrt{-5}]/(2,1+\sqrt{-5})\cong \mathbb{Z}/2\mathbb{Z}$$ The way I approached this was by using the homomorphism $f(a+b\sqrt{-5})=a+5b\: \; mod(2)$ and showing that the kernel included all multiples of 2 in the ring, as well as for $(1+\sqrt{-5})$. Is this a proper approach to proving the isomorphism? I'm sorry if this is blatantly obvious, I felt like it was a little too easy and am skeptical if whether this is a valid way of showing the isomorphism
2: $\mathbb{Z}[X]/(X^{2}+3,3)\cong (\mathbb{Z}/3\mathbb{Z})[X]/(X^2)$ For this one, I found this answer somewhere and I am confused because I ended up with something else. Like the previous example, I set up a homomorphism with a proper kernel to arrive at the isomorphism: $f(a+bx)=a+b\sqrt{-3}\; mod(3)$. This clearly has the ideal of 3 (unless I'm mistaken) and $(x^{2}+3)$ as its kernel, so would the above also be isomorphic to $(\mathbb{Z}/3\mathbb{Z})[\sqrt{-3}]$? Apart from this, how would the original isomorphism be proved?
I don't know what I'm not understanding/doing wrong. I apologize for the naivety, I'm trying to teach myself algebra using only online resources and it hasn't been very easy. Thanks for any help .