How does Mormonism (the LDS Church) answer "the problem of evil"?

Part of the "Problem of Evil" from the Church of Jesus Christ of Latter-day Saint's perspective is that following the logic of the Problem of Evil was Satan's plan before this life (cf. Moses 4:1–4—that Satan "sought to destroy the agency of man" so "that one soul shall not be lost.") This life was a test to "…prove them herewith, to see if they will do all things whatsoever the Lord their God shall command them" (Abraham 3:25). God also planned for the need for a Savior, Mediator, and Redeemer to permit all who would to overcome evil choices; Christ was therefore "prepared [for this role] from the foundation of the world" (Ether 3:14).

The only solution to the "Problem of Evil" is one in which we would have no choice but to be obedient. God, in direct opposition to His Plan, would be forced to take away our ability to be agents unto ourselves (cf. Moses 6:55–57—"And it is given unto them to know good from evil; wherefore they are agents unto themselves"— and Doctrine and Covenants 29:39—"And it must needs be that the devil should tempt the children of men, or they could not be agents unto themselves").

Alma 42 discusses the "Problem of Evil" from the perspective of why a loving and merciful God would choose to punish those who were disobedient. From the chapter synopsis: "Mortality is a probationary time to enable man to repent and serve God—The Fall brought temporal and spiritual death upon all mankind—Redemption comes through repentance—God Himself atones for the sins of the world—Mercy is for those who repent—All others are subject to God’s justice—Mercy comes because of the Atonement—Only the truly penitent are saved."

2 Nephi 2 discusses the "Problem of Evil" from the perspective of our need to have and exercise our free will—making evil our choice and not God's. From the chapter synopsis: "Redemption comes through the Holy Messiah—Freedom of choice (agency) is essential to existence and progression—Adam fell that men might be—Men are free to choose liberty and eternal life."

We also believe in the following council from God concerning our actions in this life:

26 For behold, it is not meet that I should command in all things; for he that is compelled in all things, the same is a slothful and not a wise servant; wherefore he receiveth no reward.

27 Verily I say, men should be anxiously engaged in a good cause, and do many things of their own free will, and bring to pass much righteousness;

28 For the power is in them, wherein they are agents unto themselves. And inasmuch as men do good they shall in nowise lose their reward.

29 But he that doeth not anything until he is commanded, and receiveth a commandment with doubtful heart, and keepeth it with slothfulness, the same is damned.

–Doctrine and Covenants 58:26–29

In more recent times, a talk was given in 1999 regarding Joseph Smith and the Problem of Evil. This talk is probably the best overall answer to your question in all aspects. A year earlier, an article discussing "The Richness of the Restoration" of the Gospel was published which covered this topic in some detail as well and is worthy of inclusion in this list.

From the talk, "Joseph Smith and the Problem of Evil":

For in addition to affirming that (i) God is perfectly good and (ii) all powerful, traditional Christian theologians commonly affirm two additional propositions that intensify the problem: (iii) God created all things absolutely—that is, out of nothing; and (iv) God has absolute foreknowledge of all the outcomes of His creative choices. Although apologists for belief in God have labored long to reconcile the world’s evil with God’s goodness and power, they have often overlooked the much more difficult task of reconciling evil not only with His goodness and power but with God’s absolute creation and absolute foreknowledge as well.

…

The Prophet Joseph Smith received revealed insights that do address the problem of evil in its broadest terms. His revelations suggest what might be called a soul-making theodicy, centered within a distinctively Christian soteriology (or doctrine of salvation), but both framed within a theology that rejects both absolute creation and, consequently, the philosophical definition of divine omnipotence which affirms that there are no (or no nonlogical) limits to what God can do. The Prophet’s worldview, I believe, dissolves the logical and soteriological problems of evil while infusing with meaning and hope our personal struggles with suffering, sin, and death.

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The mind of man—the immortal spirit. Where did it come from? All learned men and doctors of divinity say that God created it in the beginning; but it is not so. … I am going to tell of things more noble.

We say that God himself is a self-existent being. … [But] who told you that man did not exist in like manner upon the same principles? Man does exist upon the same principles. God made a tabernacle and put a spirit into it, and it became a living soul. … How does it read in the Hebrew? It does not say in the Hebrew that God created the spirit of man. It says, “God made man out of the earth and put into him Adam’s spirit, and so became a living body.”

The mind or the intelligence which man possesses is co-equal [co-eternal] with God himself.

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According to Lehi, there are apparently states of affair that even God, though omnipotent, cannot bring about. Man is that he might have joy, but even God cannot bring about joy without moral righteousness, moral righteousness without moral freedom, or moral freedom without an opposition in all things. With moral freedom as an essential variable in the divine equation for man, two consequences stand out saliently: (i) the inevitability of moral evil; and (ii) our need for a Redeemer.

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Earlier, when I first introduced the logical problem of evil, I argued that most discussions of the problem were too narrow and especially unfair to the Christian believer in that they failed to take into account the problem’s strongest possible solution—the incarnation of God the Son in the person of Jesus of Nazareth and his triumph over sin, suffering, and death through His atonement and resurrection. But ironically, what I referred to as “the strongest possible solution” to the problem of evil when understood in traditional terms becomes, itself, part of the problem.

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This problem that Morris and Davis state can be expressed in terms of an inconsistent triad, a set of three premises—all of which are apparently true, yet the conjunction of any two of which seemingly entails the denial of the third:

1. God is perfectly loving and just and desires that all of His children be saved.

2. Salvation comes only in and through one’s acceptance of Christ.

3. Millions of God’s children have lived and died without ever hearing of Christ or having a chance to receive salvation through Him.

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Many of you in the audience are, no doubt, smiling, recognizing that adding a premise 4 to the triad resolves the puzzle:

4. Those who live and die without having a chance to respond positively to the gospel of Jesus Christ will have that chance postmortemly.

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Confronted with what seemed to be overwhelming evil, Joseph found meaning in his suffering, maintained hope, trusted God, and kept the faith. And God spoke peace.

From "The Richness of the Restoration":

The Question of Suffering

Consider one example of a consequence of deprivation. Some, doctrinally perplexed, lament, “If God is good and all powerful, why does He permit so much human suffering? Why does He allow so much evil to be in the world He created?” A very prominent religious leader in England several decades ago spoke of this with unusual candor: “All of my life I have struggled to find the purpose of living. I have tried to answer three questions which always seemed to be fundamental: the problem of eternity; the problem of human personality; and the problem of evil. I have failed. I have solved none of them. … And I believe no one will ever solve them.” [Daily Express, London, England, 13 July 1953, 4.]

Without Restoration fulness, this problem is understandably poignant and persistent! Without the Restoration’s light on the plan of salvation, trying to comprehend this life is like trying to understand a three-act play while seeing only the second act. Without knowing beginnings and endings, the middle becomes muddled. What is really going on? Is there a director who will make sense of it all? Does the plot have purpose? Such questions are answered only by revelation.

Evil and suffering do take a terrible toll in the world, and we certainly cannot give glib answers to cover every wrenching human situation. But, through the blessings of the Restoration, we can see things as they really were, are, and will be (see D&C 93:24; Jacob 4:13). We can then better walk the straight and narrow way, inspired and informed by “faith, not by sight” (2 Cor. 5:7). However, these added understandings provided by the Restoration clearly do not exempt us from either temptation or from suffering. There are no immunities, only variations.

Latter-day Saints also know that God did not create man ex nihilo, out of nothing. The concept of an “out of nothing” creation confronts its adherents with a severe dilemma. One commentator wrote of human suffering and an “out of nothing” creation: “We cannot say that [God] would like to help but cannot: God is omnipotent. We cannot say that he would help if he only knew: God is omniscient. We cannot say that he is not responsible for the wickedness of others: God creates those others. Indeed an omnipotent, omniscient God [who creates all things absolutely—i.e., out of nothing] must be an accessory before (and during) the fact to every human misdeed; as well as being responsible for every non-moral defect in the universe.” [Antony Flew, “Theology and Falsification,” in New Essays in Philosophical Theology, ed. Antony Flew and Alasdair Macintyre (1955), 107.]