Is the polynomial $6x^4+3x^3+6x^2+2x+5\in GF(7)[x]$ irreducible? Is the polynomial $6x^4+3x^3+6x^2+2x+5\in GF(7)[x]$ irreducible?
What is the best/simplest/elementary way to approach this? Any solutions or hints are greatly appreciated.

Because the modulus is so small, we can do this the hard way.
All arithmetic will be done in $\mathbb Z_7.$
$p(x)=6x^4+3x^3+6x^2+2x+5$
$-p(x) = x^4 + 4x^3 + x^2 + 5x + 2$
$-p(x-1) = x^4 + 2x^2 + 4x + 2$
Now we suppose that
\begin{align}
   -p(x-1)
      &= (x^2 + ax + b)(x^2 + cx + d)\\
      &= x^4 + (a+c)x^3 + (ac + b + d)x^2 + (ad + bc)x + bd
\end{align}
We end up with the equations
\begin{align}
   a+c &= 0\\
   ac + b + d &= 2\\
   ad+bc &= 4\\
   bd &= 2
\end{align}
Iterating through $b$, we finally get $(a,b,c,d) = (4,1,3,2)$.
So 
\begin{align}
   -p(x-1) &= (x^2 + 4x + 1)(x^2 + 3x + 2)\\
    p(x-1) &= -(x^2 + 4x + 1)(x^2 + 3x + 2)\\
      p(x) &= -(x^2 + 5x + 3)(x^2 + 6x + 3)\\
\end{align}
