algebra precalculus - Converting $(1+...+n)^2*(n+1)^3$ to $(2+...+2n)^2$

I'm currently going through Calculus by Spivak by myself, and came across a proof by induction requiring to prove $1^3+...+n^3 = (1+...+n)^2$

Naturally, to prove this, I need to somehow convert $(1+...+n)^2+(n+1)^3$ to $(2+...+2n)^2$.
After quite a bit of thinking, I'm still not sure how to do this. I think i may be forgetting about some property of squares that we're supposed to be using.

Note: Please only provide a hint, not the complete answer.
Edit: I was mistakenly taking $(1+...+n)^2*(n+1)^3$ rather than $(1+...+n)^2+(n+1)^3$. Corrected.

Answer

Previous Answer

Hint:
$$1 + 2 + \ldots + n = \frac{n(n+1)}{2}$$

Can you take it from here?

Revised Answer

My apologies for not bothering to check more closely earlier, but I think your original equation

$$\left(1 + 2 + \ldots + n\right)^2\left(n + 1\right)^3 = \left(2 + 4 + \ldots + 2n\right)^2$$
is not an identity.

To see why, via a quick inspection, the highest power of $n$ on the LHS is $n^7$, while the highest power of $n$ on the RHS is $n^4$, a contradiction.

Therefore, your proposed equation is not an identity.