The distributive property is just one of those things that you will have to accept without a proof (called on "axiom"). The property is basically
This works even if we substitute numbers. 2(4 + 6) = 2*4 + 2*6 = 8 + 12 = 20
You can prove this by doing it another away: 2(10) = 20
Factoring is the opposite of distributing. If you have xy + xz, then you see that each term has an x in it. xy has an x and xz has an x. Then you can factor ("take out") an x because both have an x in them. xy + xz becomes x(y + z)
You can prove this with the distributive property because if you distribute the x, it becomes xy + xz again.