The numbers are 10 and -9. Since the product is negative, that implies that one number is negative and the other positive. Since the sum is 1, we know that the two numbers have to be close in magnitude. The most logical choice was 10 and -9.
Or, you could set up and solve two equations:
x + y = 1
xy = -90
Solve the first for x,
x = 1 - y
Substitute into the second,
(1 - y)y = -90
y - y² = -90
y² - y - 90 = 0
Factor
(y - 10)(y + 9) = 0
The roots are 10 and -9
If y = 10, then x = 1 - y = -9
We know that pair works.
If y = -9, then x = 1 - y = 10
That pair works, also.
