abstract algebra hw problem

Hello,

I can't seem to figure out a part of this question:

Let F be a field and let I = { a(sub n)x^n + a(sub n-1)x^(n-1) + ... + a (sub 0), where the a(sub i) are elements of F and a(sub n) + ... + a (sub 0) = 0}. Show that I is an ideal of F[x].

Here's what I have so far: So I defined an f(x) and g(x) that were elements of I and showed that f(x) - g(x) is in I. I'm trying to use the ideal test so I need to show now that, for an r(x) that's an element of F[x], r(x)f(x) and f(x)r(x) are in I. I can't figure out how to do this.

Any help is much appreciated, thanks!