Question

What two numbers multiply to -96 and add to 20

Answer 1

$xy=-96\\ x+y=20\\\\ xy=-96\\ x=20-y\\\\ (20-y)y=-96\\ 20y-y^2=-96\\ y^2-20y-96=0\\ y^2-24y+4y-96=0\\ y(y-24)+4(y-24)=0\\ (y+4)(y-24)=0\\ y=-4 \vee y=24\\\\ x=20-(-4) \vee x=20-24\\ x=24 \vee x=-4\\\\ (x,y)\in\{(24,-4),(-4,24)\}$

So these numbers are -4 and 24.

Answer 2

If they multiply to -96, that means one number will be negative. If the smaller number of the two is negative, then when you add them you will get a positive number.

Lets first find factors of -96 (just make the smaller number negative so that way when we add we can find the correct sum faster)

-96 = 

96 x -1 
48 x -2 
32 x -3
24 x -4
16 x -6
12 x -8

Now
96 + (-1) = 95
48 + (-2) = 46
32 + (-3) = 29
24 + (-4) = 20
16 + (-6) = 10
12 + (-8) = 4

The only pair that multiplies to -96 and adds to 20 is 24 and -4