How we cite our quotes: All quotations are from The Two Towers.
Quote #4
LEGOLAS: Look at them. They're frightened. I can see it in their eyes. Boe a hyn: neled herain dan caer menig! [And they should be; three hundred against ten thousand!]
ARAGORN: Si beriathar hyn ammaeg na ned Edoras. [They have more hope of defending themselves here than at Edoras.]
LEGOLAS: Aragorn, nedin dagor hen ú-'erir ortheri. Natha daged dhaer! [Aragorn, they cannot win this fight. They are all going to die!]
ARAGORN: Then I shall die as one of them!
The difference between Aragorn and Legolas isn't hope, it's acceptance of what will come to pass. Legolas despairs, unable to cope with the fact that all of these people, too old or too young to be real soldiers, are headed to their deaths. But Aragorn, knowing they have no choice, is prepared to go to his death as well. It's not that he has hope for victory, but he understands that losing faith at such a dire time will do them no good.
Quote #5
ARAGORN: What is your name?
HALETH: Haleth, son of Hama, my lord. The men are saying we will not live out the night. They say that it is hopeless.
ARAGORN: [He swings Haleth's sword and hands it back to him. This is a good sword Haleth, son of Hama. There is always hope.
Hope in The Two Towers doesn't seem to be about the likelihood of a particular outcome. Aragorn isn't saying, "Hey, we got this, don't sweat it." Instead, hope is a mindset. It's about courage in oneself and faith in your companions. That's why there is always hope, even in the bleakest of moments.
Quote #6
LEGOLAS: We have trusted you this far; you have not led us astray. Forgive me, I was wrong to despair.
Finally Legolas comes around. Notice, he doesn't say, "I decided we totally have a chance with three hundred young boys and old men against ten thousand creatures bred for sole purpose of killing all of us." His hope isn't in the odds; it's in Aragorn, his friend and leader.