abstract algebra - Order of element of multiplicative group of finite field mod polynomial

If $K$ is a finite field of size $q$ and $f$ is a degree $n$ polynomial in $K[x]$, then we can form the quotient field by modding out this polynomial. Elements of this quotient field are of the form $\phi + (f)$ where $f$ is the ideal generated by $f$. I've been trying to figure out what the order of $x^2 + (f)$ within the multiplicative group of this field, but I am struggling with how to incorporate the coefficients of this arbitrary polynomial.

I would like to understand the case where $f$ is any polynomial, but I'm struggling to even do the simpler case of when $f$ is monic and irreducible.