showing $e(n + 1) = -e(n)$, where $e(n)=f(n)^2-f(n+1)f(n-1)$ and $f(n)$ is the $n$th term of the Fibonacci sequence

Question

Let $f(n)$ be the Fibonacci sequence, and set $e(n)=f(n)^2-f(n+1)f(n-1)$ for all $n \geq 1$. Show by induction on $n \geq 1$ that $e(n+1)=−e(n)$.

I have prove the base case ($n=1$), but for the induction step, I couldn't figure out. I tried replacing $f(n)$ using recursive relation of fibonacci ($f(n+1)=f(n)+f(n-1)$), but it doesn't lead me anywhere.