Prove associative property of $\Bbb Z [x]$ ($\Bbb Z [x]$ is the set of all polynomials with variable x and integer coefficients)

Question

I can't seem to find the answer anywhere to make sure the steps I did are right or wrong, and excuse me if the question is kinda dumb :")

I want to prove that $\Bbb Z [x]$ is a ring, so I want to prove associativity on $\Bbb Z [x]$ in addition.

$\Bbb Z [x]$ is the set of all polynomials with variable x and integer coefficients with the operations of polynomial addition and multiplication.

I want to prove associativity in addition.

All I did is :

1. assuming the following: $$ƒ(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0$$ $$g(x) = b_m x^m + b_{m-1} x^{m-1} + … + b_1x + b_0$$ $$h(x) = c_s x^s + c_{s-1} x^{s-1} + … + c_1x + c_0$$ where $a_n , a_{n-1}, …, a_1, a_0, b_m , b_{m-1}, …, b_1, b_0, c_s , c_{s-1}, …, c_1, c_0 \in \Bbb Z$ and n is a non-negative integer.

2. I just said that $f(x) + (g(x) + h(x))$ and then equaled them to their values in my assumption.

That's it, that's where I don't know how I should continue to prove associativity in addition.