Quotient rings, polynomials are reducibility

Question

I am trying to follow this solution. I am struggling to understand why 'If g is a member of R, then g divides the content of f'. Why is this true?

Answer 1

If $f\in R[X]$ and $r\in R$ divides $f$, then $r$ divides every coefficient of $f$, hence the content of $f$, which is a greatest common divisor of the coefficients of $f$.

Answer 2

The content of a polynomial is the ideal generated by its coefficients. If $f= g h$ where $g$ is an element of the ring then this means that each coefficient of $f$ is a multiple of $g$, and the claim follows.