Characteristic of Field For which prime $p$, the polynomial $x^4 + x + 6$ have a root of multiplicity $&gt; 1$ over a field of characteristic $p$?

Options: 


2
3
5
7
Hint: You can check by hand all the answers
Hint:

If characteristic is 2 or 3 then the equation will be $ x(x^3 + 1 ) $
So consider $ x^3+1 $ in case of $ char \space 2 $ one is a root of $ x^3+1 $, so factor $ x^3+1 $ and check if 1 is root of factor, if not check for zero.

If above thing fails, so the same with $ Char \space3 $, here 1 is not a root of $ x^3+1 $, so only options are 2 and 0. In fact 2 is a root of $ x^3+1 = 8+1 = 0 $ so factor  $ x^3+1 $ with $ \pmod 3$ and check.

For 5 and 7 you can minimize calculation overhead by considering $ { -2, -1, 0, 1 ,2} $ and  $ { -3,-2, -1, 0, 1 ,2, 3} $