Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be irreducible over $D$ if, whenever $f(x)$ is expressed as a product $f(x)= g(x)h(x)$, with $g(x)$ and $h(x)$ from $D[x]$, then $g(x)$ or $h(x)$ is a unit in $D[x]$.
This is the definition of the irreducible polynomial in Gallian's book. What I don't understand is how this definition translate to the fact that the polynomial cannot be factored into a product of polynomials of lower degree when $D$ happens to be a field.
There is also this examples that the polynomial $f(x)=2 x^2+ 4$ is irreducible over $\mathbb R$, but reducible over $\mathbb{C}$.
To me it seems that it is also irreducible over $\mathbb{C}$ since in its factorization $2(x^2+2)$, $2$ is unit over $\mathbb{C}$.