algebra precalculus - I'm stuck on this mathematical induction problem

I've been stuck on this problem for hours, i have no idea how do even calculate it. The exponents throw me off. If anyone can help me break it down step-by-step i would truly appreciate it.
here's the problem
k= k+1

$k*2^k= 2[1+(k-1)(2^k)]$

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Answer

From your points $2$ and $3$, we need to show that

$$2(1+(k-1)2^k)+(k+1)2^{k+1}= 2(1+k2^{k+1})\\\stackrel{\text{divide by 2}}\iff 1+(k-1)2^k+(k+1)2^{k}= 1+k2^{k+1}\\\stackrel{\text{cancel out 1}}\iff (k-1)2^k+(k+1)2^{k}= k2^{k+1}\\\stackrel{\text{divide by} \,2^k}\iff (k-1)+(k+1)= 2k$$