Expansion of an expression. I want to know how to expand expressions like $(x+y+z)(a+b+c)$, I currently have a problem I want to solve but I know FOIL if it is $(x+y)(a+b)$ but what do I do when it is $(x+y+z)(a+b+c)$?
 A: Use the distributive property. 
We have that 
$$(x+y+z)(a+b+c)=x(a+b+c)+y(a+b+c)+z(a+b+c)$$ 
Can you continue? 
A: Let's create a new number $p$, which equals $y + z$. We also create $q$, which is $b + c$. Now your product becomes
$$
\begin{align}
&(x + y + z)(a + b + c) \\
&= (x + p)(a + q) \\
&= xa + xq + pa + pq \\
&= xa + x(b + c) + (y + z)a + (y + z)(b + c) \\
&= xa + xb + xc + ya + za + yb + yc + zb + zc \\
&= xa + xb + xc + ya + yb + yc + za + zb + zc
\end{align}
$$
But wait! We could have done this more easily! Just like the FOIL rule, we calculate $x(a + b + c)$, then add it to $y(a + b + c)$, then add it to $z(a + b + c)$, and we end up with the same thing: $xa + xb + xc + ya + yb + yc + za + zb + zc$. 
The important point here is that when you add a lot of things -- such as $x$, $y$, and $z$, and maybe even $w$, $v$, $u$, $\ldots$ -- and multiply the sum by some other number -- such as $(a + b + c)$ -- it is the same as multiplying each individual number by that number and then adding the results together -- for example, $x(a + b + c) + y(a + b + c) + z(a + b + c) + w(a + b + c) + \dotsc$. In symbols,
$$
(x + y + z + w + \dotsb)(a + b + c + \dotsb) \\
= x(a + b + c + \dotsb) + y(a + b + c + \dotsb) + z(a + b + c + \dotsb) + w(a + b + c + \dotsb) + \dotsb
$$
Big mathematicians call this 'distributivity'.
