<s>[INST] Proof by induction help. I seem to be stuck and my algebra is a little rusty Stuck on a homework question with mathematical induction, I just need some help factoring and am getting stuck.
$\displaystyle \sum_{1 \le j \le n} j^3 = \left[\frac{k(k+1)}{2}\right]^2$
The induction part is: $\displaystyle \left[\frac{k(k+1)}{2}\right]^2
+(k+1)^3$ is where I am having a problem. 
If you could give me some hints as to where to go since I keep getting stuck or writing the wrong equation. 
I'll get to $\displaystyle \left[{k^2+2k\over2}\right]^2 + 2{(k+1)^3\over2}$
Any push in the right direction will be appreciated. 
 [/INST] You have:
$$
\begin{align*}
\left[ \frac{n (n + 1)}{2} \right]^2 + (n + 1)^3
  &= \frac{n^2 (n + 1)^2 + 4 (n + 1)^2 (n + 1)}{4} \\
  &= \frac{(n^2 + 4 (n + 1)) (n + 1)^2}{4} \\
  &= \frac{(n^2 + 4 n + 4) (n + 1)^2}{4} \\
  &= \frac{(n + 2)^2 (n + 1)^2}{4} \\
  &= \left[\frac{(n + 1) (n + 2)}{2} \right]^2
\end{align*}
$$
</s>