Is a subring a UFD? 
If $K$ is a field is $K[X^2,X^3]$ a UFD when considered as a subring of $K[X]$?

I know that $K[X]$ is a $PID$ but nothing else ,these ring theory question just dont seem to occur to me..please help...someone told me it is not a $UFD$ and only just a Domain...but i dont know the reason..how should one go about proving these results?..this ring theory course is getting to hard for my understanding..

No, $x^2$ and $x^3$ are non-associate irreducibles in the given ring and $x^8=(x^2)^4=(x^3)^2(x^2)$.
