Great job! You might be interested in a simpler proof. One approach is to use experimental mathematics by playing around and discovering statements that seem true. If you accumulate enough then you'll find the right ones that can form lemmas to build a proof. Another approach is to look at the note in the book. Wolfram says that x^2-x*y-y^2=+-1 exactly when x and y are consecutive Fibonacci numbers. You could prove that lemma. Take a look at this expression for starters.
Simplify[(a + b)^2 - (a + b)*b - b^2]
