For Legendre's Equation $P_n(x)$: $(1-x^2)P_n^{''}(x) - 2xP_n^{'}(x) + n(n+1)P_n^{}(x) = 0 $

Question

Claim

Let $P_n: \Bbb R \mapsto \Bbb R $, $n \in \Bbb N\cup\{0\}$

$P_n(x):={1\over{2^n{n!}}}{d^n\over dx^n}[(x^2-1)^n]$, then $P_n$ has n distinct roots in ]-1, 1[

Then show that below equation is fulfilled.

$(1-x^2)P_n^{''}(x) - 2xP_n^{'}(x) + n(n+1)P_n^{}(x) = 0$

Question

I had tried brute way, but it doesn't conclude to 0. and also induction doesn't work to me too.

Any other way to show this?