Let $Q(x) = (x+1)P(x) - x$ , and clearly, $Q(x)$ has a degree of $n+1$ .
Then, for $k=0,1,2,\ldots,n$ , $Q(k) = (k+1)P(k) - k = (k+1)\cdot \dfrac{k}{k+1} - k = 0$ .
Thus, $k=0,1,2,\ldots,n$ are the roots of $Q(x)$ .
Since these are all $n+1$ of the roots of the $n+1^{\text{th}}$ degree polynomial, by the Factor Theorem , we can write $Q(x)$ as $$Q(x) = c(x)(x-1)(x-2) \cdots (x-n)$$ where $c$ is a constant.
Thus, $$(x+1)P(x) - x = c(x)(x-1)(x-2) \cdots (x-n).$$
We plug in $x = -1$ to cancel the $(x+1)P(x)$ and find $c$ :
\begin{align*} -(-1) &= c(-1)(-1-1)(-1-2) \cdots (-1-n) \\ 1 &= c(-1)^{n+1}(1)(2) \cdots (n+1) \\ c &= (-1)^{n+1}\dfrac{1}{(n+1)!} \\ \end{align*}
Finally, plugging in $x = n+1$ to find $P(n+1)$ gives:
\begin{align*} Q(n+1)&=(n+2)P(n+1)-(n+1)\\ (-1)^{n+1}\dfrac{1}{(n+1)!}\cdot(n+1)! &=(n+2)P(n+1)-(n+1)\\ (-1)^{n+1}&=(n+2)P(n+1)-(n+1)\\ (-1)^{n+1}+(n+1)&=(n+2)P(n+1)\\ P(n+1) &= \dfrac{(-1)^{n+1} + (n+1)}{n+2}\\ \end{align*}
If $n$ is even, this simplifies to $P(n+1) = \dfrac{n}{n+2}$ . If $n$ is odd, this simplifies to $P(n+1) = 1$ . $\Box$
~Edits by BakedPotato66