Question

what factors multiply to -1 but add to 1

Answer 1

Let the two factors be x and y.
Two factors that multiply to -1: $xy = -1$
Two factors that add to 1: $x + y = 1$
From the second equation: $x = 1-y$
Plug x back into the first equation: $(1-y)(y) = -1 \\ y-y^2 = -1 \\ y^2-y-1 = 0$
There are no binomial pairs that result in this quadratic. Thus we must use the quadratic formula.
$y^2-y-1=0$ 
$\frac{-b+- \sqrt{b^2-4ac} }{2a} \\ \\ \frac{1 +- \sqrt{1-4(1)(-1)} }{2} \\ \\ \frac{1 +- \sqrt{5} }{2}$

x = -y = $\frac{1 +- \sqrt{5} }{2}$
Either x or y could be the positive factor. The other variable would have to be the negative factor.

Just a sidenote: $\frac{1 +- \sqrt{5} }{2}$ is actually called the "Golden Ratio" because the ratio is found so often in beautiful example in nature.