Stuck with solving a polynomial I am doing a problem for homework that says:
Suppose $s(x)=3x^3-2$. Write the expression $\frac{s(2+x)-s(2)}{x}$ as a sum of terms, each of which is a constant times power of $x$.
I was able to do the following work for this problem:
$\frac{3(2+x)^3-3(2)^3-2}{x}$
$\frac{3(x^3+6x^2+12x+8)-24-2}{x}$
$\frac {3x^3+18x^2+36x+24-24-2}{x}$
$\frac {3x^3+18x^2+36x-2}{x}$
This is where I got stuck. I am not sure what I am supposed to do next. The multiple choice answers are:
a) $2x^2-36x+18$
b) $3x^2+18x+36$
c) $18x^2+18x+36$
d) $x^3+18x^2+36x$
e) $-3x^2-18x-36$
The closest answer to the answer I got was d, does anyone know how I would solve this?

You went astray at the first step, when you got $$\frac{3(2+x)^3-3(2)^3-2}{x}\;;$$ in fact 
$$s(2+x)-s(2)=\Big(3(2+x)^3-2\Big)-\Big(3\cdot2^3-2\Big)=3(2+x)^3-3\cdot2^3\;.$$
Can you straighten the rest out from there? When you do it correctly, there will be no constant term in the numerator.
