How to convert this algebraic equation? I am working with mathematical induction, but it gets harder when it comes to convert (or change) the form of the equation with algebra.
I have: $2+(k-1)2^{k+1} + (k+1)2^{k+1}$
And want it to reach this form: $2+((k+1)-1)2^{(k+1)+1}$
What are algebra rules/steps or simplification rules/steps I can use to reach the required form?
 A: Basic algebra:
$$\begin{align*}
(k-1)2^{k+1} + (k+1)2^{k+1} &= \Bigl( (k-1)+(k+1)\Bigr)2^{k+1} \quad\text{(distributivity of }\times\text{ over }+\text{)}\\
 &= 2k\cdot 2^{k+1} \quad\text{(performing the operation)}\\
&= k(2^12^{k+1})\quad\text{(commutativity and associativity of }\times\text{)}\\
&= k2^{1+k+1}\quad\text{(}2^a2^b=2^{a+b}\text{)}\\
&= (k+0)2^{(k+1)+1}\quad\text{(}x+0=x\text{)}\\
&= \Bigl(k+(1-1)\Bigr) 2^{(k+1)+1}\quad\text{(}a-a=0\text{)}\\
&= \Bigl( (k+1)-1\Bigr)2^{(k+1)+1}\quad\text{(associativity of }+\text{)}.
\end{align*}$$
A: If you want to verify that your equation holds, a reasonable strategy is to try to express each side as "simply" as possible. So let us work separately with the left-hand side and the right-hand side, while glancing at each looking for commonalities.
Left-hand side:  The parts $(k-1)2^{k+1}$ and $(k+1)2^{k+1}$ have a common factor $2^{k+1}$. So their sum is $(2k)2^{k+1}$, and the left-hand side is equal to $2+(2k)2^{k+1}$, which can be rewritten as $2+(k)2^{k+2}$.
Right-hand side: Just doing the arithmetic gives us $2+(k)2^{k+2}$.
