I just need a counter example for each of theses problems, they have been killing me for a few days now..

A) The gcd of

$\displaystyle 2x^2+4x+2$

$\displaystyle 4x^2 +12x+8$

in $\displaystyle \mathbb{Q}[x]$ is $\displaystyle 2x+2$

$\displaystyle 2x^2+4x+2=2(x+1)^2\,,\,4x^2+12x+8=4(x+1)(x+2)$, so the gcd is $\displaystyle x+1$ , up to a constant, and thus your answer is correct, though perhaps not the prettiest one.
B) If $\displaystyle k$ is a field in $\displaystyle p(x) \in k[x]$ is a non constant polynomial having no roots in $\displaystyle k$, then $\displaystyle p(x)$ is irreducible in $\displaystyle k[x]$

$\displaystyle (x^2+1)^5$ has no roots over $\displaystyle \mathbb{R}$, but it is terribly NOT irreducible... Tonio
These are both false, but Everything i try works for them...