Show that $\displaystyle R[x] / <x^2+1>$ is a field.

My proof so far:

Now, $\displaystyle R[x] / <x^2+1> = \{ ax+b+<x^2+1> : a,b \in R \}$

Now, [tex](1+<x^2+1>)[tex] is the unity of $\displaystyle R[x] / <x^2+1>$

But I have trouble trying to prove every element has a unit.

Let $\displaystyle (ux+v+<x^2+1>)$ be in $\displaystyle R[x] / <x^2+1>$

I need to find the inverse.

little help, please?

Thank you