Is the quotient ring $\mathbb{Q}[x]/I$ a field, where I is the ideal general by $f(x)=(x+1)(x+2)\left(x^{2}+1\right)$ and $g(x)=(x+1)^{2}(x+2)^{2}$?

Question

When a polynomial $p(x)\in\mathbb{Q}[x]$ is irreducible, then we know that a quotient ring $\mathbb{Q}[x]/(p(x))$ is a field. But if we consider an ideal $I$ generated by more than one polynomial, such as $f(x)=(x+1)(x+2)\left(x^{2}+1\right)$ and $g(x)=(x+1)^{2}(x+2)^{2}$, how do we determine if $\mathbb{Q}[x]/I$ is a field?