This rearranges to \[4n\left(\sum{\frac{x_k^2}{k}}\right)\left(\sum{kx_k^2}\right)\le(n+1)^2\left(\sum{x_k^2}\right)^2\]after homogenizing. Now see this.

Use the above form. Let $a_{i}=x_{i}$. Note that if $i\neq1,n$, replacing $a_{i}$ with 0, $a_{1}$ with $a_{1}+a_{i}/i$ and $a_{n}$ with $a_{n}+(i-1)a_{i}/i$ increases the left hand side and keeps the right hand side the same (both factors of the left hand side with $a_{i}$ will clearly go up). So it suffices to prove the claim for two variables, which is just one variable since we will have $a_{1}+a_{n}=0$. This much is trivial.