Binomial coefficient complex expression I have been trying to find the coefficient of
$a^2x^3$, in the expansion of $(a+x+c)^2(a+x+d)^2$, without success. 
I am having trouble expanding the above expression, because I can't find a way to merge them into one.
How can I solve this? Thanks

Here is a variation to determine the coefficient of $a^2x^3$ without a full expansion of the expression. It is convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ in an expression.

We obtain
  \begin{align*}
[a^2x^3]&(a+x+c)^2(a+x+d)^2\\
&=[a^2]\left([x^2](a+x+c)^2\right)\left([x^1](a+x+d)^2\right)\\
&\qquad +[a^2]\left([x^1](a+x+c)^2\right)\left([x^2](a+x+d)^2\right)\tag{1}\\
&=[a^2]\left(2(a+d)\right)+[a^2]\left(2(a+c)\right)\tag{2}\\
&=0
\end{align*}

Comment:


*

*In (1) we observe that in order to obtain the coefficient of $x^3$ one factor has to contribute $x$ and the other factor has to contribute $x^2$. There are no other possibilities to obtain a term with $x^3$.

*In (2) we select the coefficient of $x$ resp. $x^2$ according to the binomial formula
\begin{align*}
(a+x+u)^2=(a+u)^2+2(a+u)x+x^2
\end{align*}
Since there is no term with a factor $a^2$ the result is $0$.
