<s>[INST] How to get the RHS from the LHS $\sum_{k=0}^n2^kx^{k+2}=\frac{x^2-2^{n+1}x^{n+3}}{1-2x}\tag{1}$ How to get this required RHS from the given LHS.
$$\sum_{k=0}^n2^kx^{k+2}=\frac{x^2-2^{n+1}x^{n+3}}{1-2x}\tag{1}$$
This was used in a solution to the following question I asked. I couldn't understand the step and hence to understand in detail I put it as a question. 
Find the sum of the n terms of the series $2\cdot2^0+3\cdot2^1+4\cdot2^2+\dots$
 [/INST] $$
\sum_{k=0}^{n}{2^k{x}^{k+2}} = x^2\sum_{k=0}^{n}{2^k{x}^{k}}\\
\sum_{k=0}^{n}{2^k{x}^{k+2}} = x^2\sum_{k=0}^{n}{{(2x)}^{k}}\\
\sum_{k=0}^{n}{2^k{x}^{k+2}} = x^2\frac{1-{(2x)}^{n+1}}{1-2x}\\
\sum_{k=0}^{n}{2^k{x}^{k+2}} = \frac{x^2-x^2{2}^{(n+1)}{x}^{n+1}}{1-2x}\\
\sum_{k=0}^{n}{2^k{x}^{k+2}} = \frac{x^2-{2}^{n+1}{x}^{n+3}}{1-2x}
$$
I used the formula for summing geometric series:
$$
\sum_{k=0}^{n}{t^k} = \frac{1-{t}^{n+1}}{1-t}
$$
</s>