We have to solve the simultaneous equations

x^2 + y^2=10 ...(1)

x^4 + y^4=82 ...(2)

let a = x^2 and b = y^2

This gives

a + b = 10 and a^2 + b^2 = 82

a + b = 10

=> a^2 + b^2 + 2ab = 100

substitute a^2 + b^2 = 82

=> 82 + 2ab = 100

=> 2ab = 18

=> ab = 9

substitute a = 10 - b

b(10 - b) = 9

=> b^2 - 10b = -9

=> b^2 - 9b - b + 9 = 0

=> b(b - 9) - 1(b - 9) = 0

=> (b - 1)(b - 9) = 0

=> b = 1 and b = 9

For b = 1, a = 9

for b = 9, a = 1

Now b = y^2 = 1 => b = -1, + 1  and  a = x^2 = 3 , -3

y^2 = 9 => y = -3 and +3 and x^2 = 1 => x = -1 and +1

The required solutions are: (3, -1), (3 , 1), (-3 , -1), (-3, 1), (1 , -3), (1 , 3) ,(-1 , 3) and (-1 , -3)