You have the polynomial

Then by the rational roots theorem if it has any linear factors they have to be of the form

where are all of the factors of p and q. so the possible zeros are

If you check these in they are not zero, it does not have any linear factors.

Now comes the harder work (it isn't too bad )

If it has a non trivial factorization it must have two quadratic factors. Since the leading coefficent is 1 and 2 only has two factors it would have to look like

If we expand the left hand side we get

This gives us 3 equations that must hold

The first equation and the third equation force but this contradicts the 2nd equation. So the equation does not factor