Question

What two numbers multiply to 10 and add to -29 ?

Answer 1

There are none. 
You can prove this by graphing. 
xy = 10
y = 10/x

x + y = -29
y = -x - 29

On the graphing calculator, type in
Y1 = 10/x
Y2 = -x - 29

As you can see the two do not intersect.

Answer 2

Xy=10
x+y=29

subtract x from both sides for second equation

y=-x-29
subsitute in second equaiton
x(-x-29)=10
-x^2-29x=10
add x^2+29x to both sides
x^2+29x+10=0
quadratic formula
if you have
ax^2+bx+c=0,
x=$\frac{-b+/- \sqrt{b^{2}-4ac} }{2a}$

so
x^2+29x+10=0
a=1
b=29
c=10
x=$\frac{-29+/- \sqrt{(-29)^{2}-4(1)(10)} }{2(1)}$
x= $\frac{-29+/- \sqrt{841-40} }{2}$
x= $\frac{-29+/- \sqrt{801} }{2}$
x= $\frac{-29+/- 3\sqrt{89} }{2}$

x= $\frac{-29- 3\sqrt{89} }{2}$ or $\frac{-29+ 3\sqrt{89} }{2}$
those are the 2 numbers
aprox=-28.65 and -0.349