This comes out of a series of transformations, so I'll just get to the main focus here.

Define the functions $$F_n(x)=\frac12\left(x+2+2\sqrt{x+1}\right)^n+\frac12\left(x+2-2\sqrt{x+1}\right)^n. \tag1$$ It's straightforward to obtain $$F_n(x)=\sum_{r=0}^n\binom{2n}{2r}(x+1)^r. \tag2$$

QUESTION 1.On the other hand, how does one obtain (3) by manipulating (1)? $$F_n(x)=\sum_{r=0}^n\frac{n}{n+r}\binom{n+r}{2r}4^rx^{n-r}. \tag3$$

QUESTION 2.As an aside, I'd also be pleased with a combinatorial proof of the sums (2)=(3).

**POSTSCRIPT.** Question 1 has received ample response and thanks for those who did. I hope someone can address Question 2.