If x^2-x-1 divides ax^6+bx^5+1 evenly, find the sum of a and b. Show all steps and exlain your answer.

I did a very messy long division and my remainder was

(8a+5b)x + 1+5a+3b
but I know this had to be zero, so
8a+5b=0 and 1+5a+3b=0

solving these two I got a=-5 and b=8

(I actually repeated the long division with those two values subbed in, and I did get a remainder of zero, thus verifying my answer)

so a+b = 3

BTW, the solution to your divisor = 0 is the Golden Ratio, which comes from the Fibonacci numbers
and the numbers 3,5, and 8 happen to be consecutive Fibonacci numbers)
probably not a coincidence, who knows.

Just in case anybody is interested. lol