Question 1145439
x² + x = 1 , is the same as x² + x - 1 = 0, it is a quadratic so you can solve for x using the quadratic formula 
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
a = 1, b = 1, c = -1
which will end up being
x = - (1 +- sqrt(5)) / 2

+- results in two answers

substitute - (1 + sqrt(5)) / 2 into x in the problem (x^5  + 8) / (x + 1) = 5
substitute - (1 - sqrt(5)) / 2 into x in the problem (x^5  + 8) / (x + 1) = 5

in this case if you plug in the two different x given by the quadratic formula you only end up with x = 5 

Answer : x = 5