We have the ring $R=\mathbb{Z}_5[x]$, the element $a=x^2+3$ of $R$ and the ideal $I=(a)$ of $R$.
If $f_k(x)=12x^2+k$, ($k\in \mathbb{Z}$), $R_1=R/I$ and the element $t$ of $R_1$ is the equivalence class of $x$, then $f_k(t)=1$ in $R_1$ for which $k$ ?
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We have that $t\equiv x \pmod 5$.
We also have that $f_k(t)=1\Rightarrow 12t^2+k=1 \Rightarrow 12t^2+k-1=0$.
So, we have to check for which $k$ the remainder of the division of $12x^2+k-1$ and $x^2+3$ is equal to $0$, or not?