$$ \sum_{n=0}^{k+1}\frac{3F_{n+1}-L_{n+1}}{2n!}\frac{(k+1)!}{(k-n+1)!}x^{k-n+1}=(\varphi+x)^k\left(\frac{\sqrt{5}}{5}-\frac{\sqrt{5}-5}{10}x\right)+(\psi+x)^k\left(\frac{\sqrt{5}+5}{10}x-\frac{\sqrt{5}}{5}\right).$$

Wolfram Alpha gave a closed form and after a bit of simplification I got the result above. How would you prove it?