How to go about computing an ideal in a polynomial ring

Question

I am stuck on how to characterize the elements in $(5,x+7)$ in the polynomial ring $\mathbb{Z}[x]$. I seriously can't even figure out if $x$ is in this ideal. I know that elements in the ideal look like $5h(x)+(x+7)g(x)$ for $h(x),g(x)\in \mathbb{Z}[x]$, and then I tried to expand this out, but I got some huge mess that I didn't know how to work with. In particular I am trying to figure out whether this ideal is maximal or prime or neither, and I thought the best way to do that was to find some elements that are not in the ideal so I could maybe find some zero divisors in $\mathbb{Z}[x]/(5,x+7)$.

Answer 1

I think your best bet is to read @Servaes answer on this post:

Is the ideal $(2,X+1)\subset\Bbb{Z}[X]$ prime, maximal or neither?

Then convince yourself that this quotient ring is a field (that you've probably seen before) and conclude that your ideal is maximal.