Let $R = k[a,b,c]$ where $k$ is a field and let $I$ be the ideal of $R$ generated by the polynomial $ac - (b^2 + 1).$ I am considering the quotient ring $$R / I = k[a,b,c] / (ac - (b^2 + 1)).$$ I want to show that this quotient ring is an integral domain.
My first thought is to show that $I$ is a prime ideal of $R$ and then one can conclude that $R/I$ is an integral domain. I am having trouble formulating an argument as to why $I$ is prime in $R$. Any suggestions on how to begin would be appreciated.