Proof by induction that $(1^3 + 2^3 + 3^3+\cdots+n^3) = (1 + 2 + \cdots + n)^2$ I'm sitting with the proof in front of me, but I do not understand it.
$$A = \{n \in Z^{++} \mid (1^3 + 2^3 + 3^3+\cdots+n^3) = (1 + 2 + \cdots + n)^2\}$$
The first step of proof by induction is simple enough,to prove that $1 \in A$
$1^3 = 1^2$
The next step is where I get tripped up. So I add $n + 1$ to the right hand side
$$(1 + 2 + \cdots + n + (n + 1))^2 = (1 + 2 + \cdots + n)^2 + 2(1 + 2 + \cdots + n)(n + 1) + (n + 1)^2$$
My algebra is failing me here, because I do not understand how the equation was expanded. 

For the inductive step:
$$(\underbrace{1+2\cdots+n}_{=A=\frac{n(n+1)}{2}}+(n+1))^2=\underbrace{A^2}_{=1^3+2^3+\cdots+n^3}+\underbrace{(n+1)^2+n(n+1)^2}_{=(n+1)^3}$$
