Zero Factor Property
Key Questions

You use the zero factor property after you have factored the quadratic to find the solutions.
It is best to look at an example:
#x^2+x6=0# This factors into:
#(x+3)(x2)=0# We find our solutions by setting each factor to zero and solve:
#x+3=0#
#x=3# or
#x2=0#
#x=2# Previous answer (I was thinking some more complicated before):
You are not using the words precisely. You use the factor theorem with the factor property. The factor theorem states that if you find a
#k# such that#P(k)=0# , then#xk# is a factor of the polynomial. The factor property states that#k# must a factor of the constant term in#P(x)# .Having said all that, you wouldn't normally use the factor theorem or factor property to solve a quadratic; they are many used to find factors of higher order polynomials. Once you reduce the higher order polynomial to a quadratic, you use regular factoring methods such as FPS or PFS: Factors, Product, and Sum.
#P(x)=ax^2+bx+c# The problem with the factor theorem and factor property is that it's not as easy to use when
#a!=1# . 
Answer:
Well, if you a polynomial is factorable then its roots/zeroes can be easily found by setting it to zero and using the zero factor property. Please see explanation below.
Explanation:
The Zero Product Property:
A product of factors is zero if and only if one or more of the factors is zero. Or:
if#a*b = 0# , then either#a = 0# or#b = 0# or both.
Example: Find the roots of the polynomial by factoring:
#P(x) = x^3x^2x+1# , set to zero:
#x^3x^2x+1=0# , factor by grouping:
#x^2(x1)1(x1)=0#
#(x^21)(x1)=0# , use difference of squares to factor further:
#(x+1)(x1)(x1)=0# , use the zero factor property:
#x+1=0=>x=1#
#x1=0=>x=1#
Notice that#x = 1# has a multiplicity of 2. 
The zero factor property states that if
#ab=0# , then either#a=0# or#b=0# .Example: find the roots of
#x^2x6# .#x^2x6=0# #(x3)(x+2)=0# Now, the zero factor property can be applied, since two thing are being multiplied and equal zero.
We know that either
#x3=0color(white)(ssss)# or#color(white)(ssss)x+2=0# Solve both to find that
#x=3# or#x=2# .