First up, I think euthenasia is immoral.
If I were writing your paper, I would first clarify what that can mean. I see 2 ways:
1. Is the PRACTICE of euthenasia immoral?
ie, is it OK to euthenise someone or to try to get euthenised?
Is it a bad thing when it happens?
2. Should we (society) allow it to happen?
ie, should it be legal?
Second one first: Should it be legal?
The reason I think this is a crucial distinction to make is that allowing it to happen leagally, whether you are sympathetic to euthenasia or not, opens up the very great possibility that the law will be abused and murder will become very easy to get away with. So although this aspect seems like purely a legal issue, I think it is really a moral/ethical question whether we should put our trust in any legal system to determine issues such as whether someone's wish to die is justifiable Eg, do they really have a terminal illness? Are they just depressed and suicidal? What about cases where other parties will benefit (financially, etc) from the death?
It seems to me that legalising it would put a lot of pressure on doctors to make the right legal call, then go through with the euthanising, then defend any challenges to the legality of the act in court after the fact, under risk of being charged with murder. If you legally take away the risk of a murder charge, so that doctors found to be acting outside the law cannot be charged with murder and only charged with breaking euthenasia laws, then it becomes a very inviting avenue for people who wish to commit murder.
Now the first: Is the PRACTICE of euthenasia immoral?
I think it's not OK. Killing is bad, that's uncontroversial. Suicide is illegal in most places. I see euthenasia as a special category of suicide, where it's justification is that the pain and suffering of people with terminal or chronic illnesses (ie, they'll never feel better) have a right to die, or that someone else can make that decision for them, because it is the only way to end their suffering. Most suicides are commited by depressed people, who feel so much pain and suffering that they kill themselves. The reason we don't generally allow suicide is that the pain and suffering can be alleviated, the person can be helped out of the dead end (pun!) they've found themselves in. With the high quality of palliative care that is available in most countries these days, the same is true for all pain and suffering. There really is no need to suffer pain. Death is not the only option.
Personally, I would be against any kind of suicide, even when it means eduring unbearable agony. But I don't think anyone has a right to impose this view on others. Explain why it is wrong, hope that others will be convinced to stay alive to experience the pain. After all, how many of us want to live simply to experience pain-free comfort and physical pleasure. There is more to life than that. If you could have an operation that would mean you could only laugh and never, ever cry, would you have it done? Not me. We are all spiritual creatures, even though it often seems some people are not. I get very upset when I hear euthenasia supporters say that we should have the right to die with dignity. I don't see it as dignified at all! Wanting to stay alive, despite the agony you are in, that's dignified to me.
PS I don't believe there is an afterlife. I believe that when you die, that's it, everything ends. But even if you do believe there is something after death, there is no way of determining and knowing for sure, there is no reason to believe it. So the notion that euthanasia will end earthly life only and then allow you to move on to some other afterlife, doesn't hold any weight morally.
PPS I live in Australia, where recently there was a widely publicised case of a cancer patient that wanted to die. It is illegal here. After a few weeks of media attention, she went through with it (she bought the lethal drugs overseas via the internet). About a week later it turns out she was not dying, her cancer was in remission and she was physically doing OK, and definitely not in unbearable pain. Many had claimed, before she died, that she was mentally unstable. And it turns out now that was true, the implication being that she was exploited by euthenasia campaigners in the most immoral way. It's also an ideal case study for supporting an anti-euthenasia position. The woman's name was Nancy Crick, she died in the state of Queensland in Australia this year (April or May?), if you want to look it up on the net.