First of all sorry for the stupid question but my knowledge in mathematics is not big. I am studying by myself how determine the existence condition for a polynomial. In particular I am not sure to perform correctly the following exercise.
My book says: determine the the existence condition of the following expression before simplifying this expression $$\left(\frac{a^2}{4a^2+4ab+b^2}-\frac{a-b}{6a+3b}\right):\frac{a^3-b^3}{12a+6b}$$ First of all I would rewrite the expression in this way: $$\frac{\left(\frac{a^2}{\color{red}{(2a+b)^2}}-\frac{a-b}{\color{green}{3(2a+b)}}\right)}{\frac{a^3-b^3}{\color{blue}{6(2a+b)}}}$$ so since I should have all denominators in the expression different from $0$, I have imposed the following: $$\begin{cases} \color{red}{(2a+b)^2}\neq 0\\ \color{green}{3(2a+b)}\neq 0\\ \color{blue}{6(2a+b)}\neq 0\\ a^3-b^3\neq 0\\ \end{cases}\iff b\neq -2a; a\neq b $$
$\textbf{My doubt:}$ I am not sure on what I have done. In particular I have imposed, in addition to the first three condition (red, blue and green) referred to the denominator of the fractions in the expression, also that the term $\frac{a^3-b^3}{\color{blue}{6(2a+b)}}$ has to be different from $0$.
I have thought that since it is a factor that the exercise tell us to divide thanks to the symbol "$:$", in addition to the existence condition on this fraction I should be sure that this fraction is not identically $0$ since it represents a denominator for the term $\left(\frac{a^2}{4a^2+4ab+b^2}-\frac{a-b}{6a+3b}\right)$: so I have to impose also the condition in black, right?
Can you help me please?
Everything that you have done is correct, though I'm going to add some extra notes here as you seem a little uncertain and say that you're teaching yourself.
You're looking at what are called rational polynomials as they are the ratio of polynomials (or "fractions") and so the existence condition is basically that these things are not allowed to be undefined. That happens if you divide by $0$, and so that is why you're checking the conditions that you're checking.
The exercise is first to determine if each of the polynomial fractions exists, and then to determine if their ratio (indicated by the ":" as you noted) exists, which again requires division as you have understood.
Everything else is fine; you can simplify your calculations a little by noticing that the term $(a+2b)$ is present and dominant in the red, green and blue terms, so all three are controlled by requiring $a\not=2b$.
Finally, yes, you need to be sure that $a^3 - b^3 \not= 0$ as well for the final ratio/division check. This is the first time when it might matter what number field you're working in: $a^3-b^3 = (a-b)(a^2+2ab+b^2)$ and so if you're working over $\mathbb C$ then there will be two additional complex roots that $a$ and $b$ are not allowed to take on.