$\begin{align*}
(1+\dots+(n+1))^2 &= (1+\dots+n)^2 + 2(1+\dots+n)(n+1)+(n+1)^2 \\
&= 1^3+\dots+n^3 + 2(1+\dots+n)(n+1)+(n+1)^2 \\
&= 1^3+\dots+n^3 + n(n+1)^2+(n+1)^2 \\
&= 1^3+\dots+n^3 + (n+1)^3
\end{align*}$