Prove the given property of the Fibonacci numbers directly from the definition. $$
[F(n + 1)]^2 = [F(n)]^2 + F(n − 1)F(n + 2) 
\text{ for } n \geq 2
$$
I started by breaking up the leftmost term into:
$$
    F[n+1] \cdot F[n+1]
$$
I then used the Fibonacci relation to break them into:
$$
    [F(n) + F(n-1)][F(n) + F(n-1)]
$$
I multiplied them together to give myself:
$$
    [F(n)]^2 + 2[F(n-1)F(n)] + [F(n-1)]^2
$$
Though I am stuck after this part. If anyone can offer me insight as to whether I am heading in the right direction, or point me in the right direction, I would really appreciate it!

Note that the largest index that appears in the identity is $n+2$, so you want to apply the recurrence there:
$$
F(n+2) = F(n+1) + F(n).
$$
Can you take it from there?
