$\begin{align*}
F_{n+1} &= F_n + F_{n-1} \\
&= \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^n\sqrt{5}} + \frac{(1+\sqrt{5})^{n-1} - (1-\sqrt{5})^{n-1}}{2^{n-1}\sqrt{5}} \\
&= \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^n\sqrt{5}} + \frac{2(1+\sqrt{5})^{n-1} - 2(1-\sqrt{5})^{n-1}}{2^n\sqrt{5}} \\
&= \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n+2(1+\sqrt{5})^{n-1} - 2(1-\sqrt{5})^{n-1}}{2^n\sqrt{5}} \\
&= \frac{(1+\sqrt{5})^{n-1}((1+\sqrt{5})+2) - (1-\sqrt{5})^{n-1}((1-\sqrt{5})+2)}{2^n\sqrt{5}} \\
&= \frac{(1+\sqrt{5})^{n-1}(3+\sqrt{5}) - (1-\sqrt{5})^{n-1}(3-\sqrt{5})}{2^n\sqrt{5}}
\end{align*}$