Binomial theorem question (How did we get the following equality?)

Question

I'm not clear as to how we get the following equality:

$$\dfrac{(x+1)^p - 1}{x} = x^{p-1} + {p \choose 1}x^{p-2} + \cdots + {p \choose p-2}x + {p \choose p-1}. $$

Answer 1

If $p$ is a positive integer then

$$\begin{split}(x+1)^p&=\sum_{i=0}^p{p\choose i}x^i1^{p-i}\\ &={p\choose 0}x^p+{p \choose 1}x^{p-1}+...+{p\choose p-1}x^1+{p\choose p}x^0\end{split}$$

${p \choose p}x^0=1$ so subtracting $1$ from the expression removes it. Afterward, division by $x$ reduces all the powers by $1$. So

$$\frac{(x+1)^p-1}x=x^{p-1}+{p \choose 1}x^{p-2}+...+{p\choose p-1}$$