let 'x' and 'y' be the numbers

the sum of the squares of two numbers is 10

x^2 + y^2 = 10

the sum of the two numbers is -2

x + y = -2 => y = -2 - x

plug y = -2 - x into x^2 + y^2 = 10

x^2 + (-2 - x)^2 = 10

by solving the quadratic equation we find the roots

x1 = 1

x2 = -3

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thus we have 2 solutions

x=1 and y = -2 - 1 = -3

x = -3 and y = -2 -(- 3) = 1