What is the zero product property and how do you use it?

Anything multiplied by zero is zero. And its converse, that the product of non-zero numbers must be non-zero.

This is used to find the roots of equations.

If a function, f(x), is equal to the product of functions g(x) and h(x),

that is, f(x) = g(x)*h(x)

then

f(x) = 0 implies that g(x) = 0 or h(x) = 0

At high school level, this is used to find the solutions of quadratic equations:

If (x - a)*(x - b) = 0 then x - a = 0 or x - b = 0

that is, x = a or x = b