2022/020) Given $m^2 = n +2$, $m^2= n + 2$ and m and n are unequal Compute $4mn - m^3 - n^3$

we are given 

$m^2 = n +2 \cdots(1)$

$n^2 = m + 2\cdots(2)$

$4 mn - m^3 - n^3$

$= 4mn - m(m^2)  - n(n^2)$

$= 4mn - m(n+2)  - m(n+2)$ using (1) and (2)

$= 2mn - 2(m+n)$

or $4 mn - m^3 - n^3=  2mn - 2(m+n)\cdots(3)$ 

we need to commte mn and m + n

sutarcting (2) from (1)

$m^2 - n^2 = n - m$

as n-m is not zero divding by n-m we have

$m+n = -1\cdots(4)$

adding (1) and (2)

$m^2 + n^2 = (m+n) + 4 = -1 + 4 = 3\cdots(5)$ putting value of m + n from (4)

or

Hence $2mn = (m+n)^2 - (m^2+n^2) =  1- 3 = -2$

putting the values from (4) and (6) in (3) we get  

 $4 mn - m^3 - n^3=  2mn - 2(m+n) =  -2 -2 (-1) = 0$