Decide whether the following polynomials are reducible or not in the given ring.
$f(x)=5x^4+22x^3+35x^2+47x+15$ in Q$[x]$
$g(x)=2x^4-4x^3-12x^2+14x-2$ in Q$[x]$
I am not sure what method to use to determine irreducibility here. I know, for quadratic and cubic polynomials, the polynomial is irreducible if it has no roots in Q$[x]$. But of course these are quartic polynomials. I cannot use the Eisenstein Criterion because no prime number divides the coefficients of $f(x)$ and the only prime number, 2, that divides the coefficients of $g(x)$ also divides the leading coefficient, which is not acceptable when using the Criterion. I would also want to convert the coefficients of each to modulo 2 so that I could show they are irreducible in F$_2$[x], but this only works for monic functions. Any idea of what I can do?