  Solving the equation $P_a(X):=X^{q+1}+X+a=0$ over finite field $\GF{Q}$,
where $Q=p^n, q=p^k$ and $p$ is a prime, arises in many different contexts
including finite geometry, the inverse Galois problem \cite{ACZ2000}, the
construction of difference sets with Singer parameters \cite{DD2004},
determining cross-correlation between $m$-sequences
\cite{DOBBERTIN2006,HELLESETH2008} and to construct error-correcting codes
\cite{Bracken2009}, as well as to speed up the index calculus method for
computing discrete logarithms on finite fields \cite{GGGZ2013,GGGZ2013+} and on
algebraic curves \cite{M2014}.
  Subsequently, in
\cite{Bluher2004,HK2008,HK2010,BTT2014,Bluher2016,KM2019,CMPZ2019,MS2019}, the
$\GF{Q}$-zeros of $P_a(X)$ have been studied: in \cite{Bluher2004} it was shown
that the possible values of the number of the zeros that $P_a(X)$ has in
$\GF{Q}$ is $0$, $1$, $2$ or $p^{\gcd(n, k)}+1$.
  Some criteria for the number of the $\GF{Q}$-zeros of $P_a(x)$ were found in
\cite{HK2008,HK2010,BTT2014,KM2019,MS2019}. However, while the ultimate goal is
to identify all the $\GF{Q}$-zeros, even in the case $p=2$, it was solved only
under the condition $\gcd(n, k)=1$ \cite{KM2019}.
  We discuss this equation without any restriction on $p$ and $\gcd(n,k)$. New
criteria for the number of the $\GF{Q}$-zeros of $P_a(x)$ are proved. For the
cases of one or two $\GF{Q}$-zeros, we provide explicit expressions for these
rational zeros in terms of $a$. For the case of $p^{\gcd(n, k)}+1$ rational
zeros, we provide a parametrization of such $a$'s and express the $p^{\gcd(n,
k)}+1$ rational zeros by using that parametrization.
