ab = 1 a + b = -4 → b = -4 - a
Subbing the last equation into the first, we have
a[-4 - a] = 1 simplify
-4a - a^2 = 1 rearrange
a^2 + 4a + 1 = 0 using the quadratic formula to solve, we have
a = √3 - 2 or a = -√3 - 2
Which means that b = -4 - [ √3 - 2] = -2 - √3 or b = -4 - [ -√3 - 2 ] = √3 - 2
So
(a^2 - b) * (b^2 - a) = [ ( √3 - 2)^2 - ( -2 - √3) ] * [ ( -2 - √3)^2 - (√3 - 2) ] =
[ 3 -4√3 + 4 + 2 + √3] * [ 4 + 4√3 + 3 - √3 + 2] =
[ 9 - 3√3] * [ 9 + 3√3 ] =
81 - 27 =
= 54
Evaluating this when a = -√3 - 2 and b = √3 - 2 gives exactly the same result as shown below
(a^2 - b) * (b^2 - a) = [ ( -√3 - 2)^2 - ( √3 - 2) ] * [ ( √3 - 2 )^2 - (-√3 - 2) ] = 54