Proving Boolean algebra equations is just like proving other equations. You should start with one side, and, by a chain of equations, reach the other side.
(x+y)(x+z) = xx+xz+xy+yz = x + xz +xy +yz = x(1 + z +y) + yz = x1+yz = x + yz
(x+y)(x+y!) = xx +xy +xy! +yy! = x + xy +xy! + 0 = x(1+y+y!) = x1 = x