Are $k[x^2,x^3+x]$ and $k[x^3,x^2+x]$ UFD's?

Question

It is well-known that $R=k[x^2,x^3]$ is not a UFD, since $x^2x^2x^2=x^3x^3$ are two different decompositions of $x^6$ to irreducibles.

Question 1: Are $A=k[x^2,x^3+x]$ and $B=k[x^3,x^2+x]$ UFD's?

It seems to me that they are, but I have not found a proof (or a counterexample).

Question 2: If $A$ and $B$ are UFD's, is it because one of the generators ($x^3+x$ or $x^2+x$) is separable (=have different roots)? In $R$ both generators are not separable.

Thank you very much!

Answer 1

In $A$, write $f = x^2, g = x^3 + x$. Then we have

$$g^2 = (x^3 + x)^2 = x^2 (x^2 + 1)^2 = f(f + 1)^2.$$

In $B$, similarly write $f = x^3, g = x^2 + x$. Then we have

$$g^3 = (x^2 + x)^3 = x^3 (x^3 + 3x^2 + 3x + 1) = f(f + 3g + 1).$$

Probably a more geometric approach is possible that would give some conditions under which $k[f, g]$ isn't a UFD.