A "mormon-bashing" friend says that we can't have prophets because they've changed their minds--polygamy, blacks and the priesthood. How do you answer that question to a non-member?
Let's begin by considering the words of the prophet Joseph Smith:
"Happiness is the object and design of our existence; and will be the end thereof, if we pursue the path that leads to it; and this path is virtue, uprightness, faithfulness, holiness, and keeping all the commandments of God. But we cannot keep all the commandments without first knowing them, and we cannot expect to know all, or more than we now know unless we comply with or keep those we have already received. That which is wrong under one circumstance, may be, and often is, right under another.
God said, 'Thou shalt not kill;' at another time He said, 'Thou shalt utterly destroy.' This is the principle on which the government of heaven is conducted--by revelation adapted to the circumstances in which the children of the kingdom are placed. Whatever God requires is right, no matter what it is, although we may not see the reason thereof till long after the events transpire. ...
A parent may whip a child, and justly, too, because he stole an apple; whereas if the child had asked for the apple, and the parent had given it, the child would have eaten it with a better appetite; there would have been no stripes; all the pleasure of the apple would have been secured, all the misery of stealing lost. ... This principle will justly apply to all of God's dealings with His children. Everything that God gives us is lawful and right ..." (Teachings of the Prophet Joseph Smith, p.256)
So, this is why you and I taught on our respective missions that people needed first to gain a testimony of the message of restoration. Regardless of how much we dislike the idea, we must understand that not all questions are going to be answered in this life. Why polygamy? Why the priesthood ban? Why sudden and painful infant deaths? Why long debilitating terminal illnesses? Why can't I have today the blessings I desperately need and fervently pray for--including blessings promised me years ago? Why did my neighbor--who appears to not even understand the meaning of the same--receive the blessings I expected for myself?
The young women have a song titled (if I remember well in English) "I Walk by Faith." Sometimes I remember the lyrics (in Portuguese, of course) of that song. The reward we aim to receive is infinite; and the challenges, trials, and tribulations ahead of us will refine us to the point that we will be able to inherit everything through Jesus Christ.
"For all things are for your sakes, that the abundant grace might through the thanksgiving of many redound to the glory of God. For which cause we faint not; but though our outward man perish, yet the inward man is renewed day by day. For our light affliction, which is but for a moment, worketh for us a far more exceeding and eternal weight of glory; While we look not at the things which are seen, but at the things which are not seen: for the things which are seen are temporal; but the things which are not seen are eternal." (2 Corinthians 4:15-18)President John Taylor taught the following:
"... it is necessary that we pass through certain ordeals, and that we be tried. But why is it that we should be tried? There is just the same necessity for it now that there was in former times. I heard the Prophet Joseph say, in speaking to the Twelve on one occasion: 'You will have all kinds of trials to pass through. And it is quite as necessary for you to be tried as it was for Abraham and other men of God, and (said he) God will feel after you, and He will take hold of you and wrench your very heart strings, and if you cannot stand it you will not be fit for an inheritance in the Celestial Kingdom of God.' ... God tries people according to the position they occupy. ... But all these personal things amount to but very little. It is the crowns, principalities, the powers, the thrones, the dominions, and the associations with the Gods that we are after, and we are here to prepare ourselves for these things. We are after eternal exaltation in the Celestial Kingdom of God. And we want to feel that this is the main object of existence, that this is why we were born ..." (Journal of Discourses 24:197 - June 18, 1883)After all these considerations, let me address your original question: how to explain these things to a non-member? My answer (you probably have guessed) is simple: I don't. I just help them gain a testimony--and if they can't obtain a testimony by the power of the Spirit, not even an explanation made by an angel from on high would do them any good. Just remember how Laman and Lemuel returned to their doubts and complaints immediately after an angel had told them that the Lord would deliver Laban into their hands (1 Nephi 3:29-31).