What are two numbers that multiply to get to 4 and add to - 5?

Let the two numbers be represented by x and y. Then xy = 4 and x + y = - 5.

Solving the first equation for y, we get y = 4/x.

Subbing 4/x for y in the second equation, we get x + 4/x = - 5.

Multiplying all three terms by x removes the fractional term:

x^2 + 4 = - 5x

Rearranging terms in descending powers of x: x^2 + 5x + 4 = 0

Factoring, (x+1) (x+4) = 0, so that x = - 1 and x = - 4 are roots / solutions.

Note that (-1) (-4) = 4 and that (-1) + (-4) = - 5, as required.

Two such numbers are {-1, - 4).