If our two numbers are x and y, we can write two equations.
x + y = -10
xy = 24
Then, we can solve.
x = -10 - y
(-10 - y) * y = 24
-10y - y^2 = 24
0 = y^2 + 10y + 24
0 = y^2 + 6y + 4y + 24
0 = y (y + 6) + 4 (y + 6)
0 = (y + 6) (y + 4)
So, y can be either -6 or -4. Now, notice that if y is -6, x is -4, and if y is -4, x is -6. This is because there were no provided discrepancies between the two numbers.
Anyways, our numbers will be -6 and -4.