> 
> Let $f (x)= x^7 - 105 x +12$ then
> 
> 
> 1. $f (x)$ is reducible over $ \mathbb {Q} $
> 2. There exists an integer $m$ such that $f (m)=105$
> 3. There exists an integer $m$ such that $f (m)=2$
> 4. $f (m)$ is not a prime number for any integer $m$
> 
> 
> 

By Eisenstein 1 option is false. For option 2, if possible let there be an integer such that $f (m)=105$ then

$ m^7 - 105 m +12= 105$ implies $ m^7 -105 m -93=0$, $m$ can be $1, -1, 3, -3, 31, -31, 93, -93$ so clearly option 2 is false and in a similar way option 3 is also false. I have no idea about option 4.

So my request is to check whether my reasoning of option 1, 2 and 3 are alright and to give me a hint for option 4.

Thank you in advance.