Im trying to understand this concept in general but as an example lets say I have a polynomial like: $$f=(1+x+x^2)(1+x+2+(x+2)^2+(x+2)^3+(x+2)^4)(1+x+6+(x+6)^2 $$ which can simplify to:
$$ f= (x^8 + 23 x^7 + 214 x^6 + 1052 x^5 + 3031 x^4 + 5350 x^3 + 5844 x^2 + 3843 x + 1333) $$
and I have another polynomial based on it say: $$g=2x^2(x+2)^4(x+6)^2 $$ where I took the largest part of the components of f. Then $$g=2 x^8 + 40 x^7 + 312 x^6 + 1216 x^5 + 2528 x^4 + 2688 x^3 + 1152 x^2 $$ (1) What I am confused about is why the coefficients of f increase up until the $x^2$ term and why the coefficients of g only increase up until the $x^3$ term?
(2) If I wanted to show that term by term the rates of how the coefficients are increasing up until they decrease is that possible? My thought for example on g was to try something like : $$\dfrac{1216-312}{312-40}$$
(3) If anyone can suggest how to generalize any answers to (1) and (2)