Show that $Z^2 + Y^3 + X^5$ is irreducible in $\mathbb C[X,Y,Z].$

Question

Here is the question I want to answer:

Let $\mathbb C[X,Y,Z] \cong \mathbb C^{[3]}.$ Define rings $$ A = \mathbb C[Y,Z]/(Z^2 + Y^3) \text{ and } B = \mathbb C[X,Y,Z]/(Z^2 + Y^3 + X^5) = \mathbb C [x,y,z]$$ where $x,y,z$ are the images of $X,Y,Z$ under the standard projection $\mathbb C [X,Y,Z] \rightarrow B.$

$(b)$ Show that $Z^2 + Y^3 + X^5$ is irreducible in $\mathbb C[X,Y,Z].$ Conclude that $B$ is an integral domain.

Here is my trial:

1- Can anyone give me a feedback on my trial please? specifically,I feel like my reasoning that it is an integral domain is not correct.

Could anyone show me how to find this isomorphism and its kernel and image please?

EDIT:

3-I also found this question here:

Show that $f(x,y,z)=x^2-y^2z$ is irreducible in $\mathbb{C}[x,y,z]$. but still I am confused about the general procedure and the specific details I should calculate to solve those kinds of problems, could anyone clarify this to me please?