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
l*(m+n) = lm + ln

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.

I hope I helped at least a little.