Talk to the two-headed dragon. use the compass to find the scale and approach it from the left. Slash the tree with your sword and you can get at it. Use the scale to jump from rock to rock. Find and talk to the swordsman. Go into the cave on the right edge of the screen.
Kill everything you can, and take everything you can. Move to the furthest corner up and left. use the compass. Your destination is the room at the bottom right of the displayed map. (Without scrolling) Go in the room, talk to the dragon, then exit the room and the cave. Go right from the cave to the next screen for the last egg, then return to the dragon.
Talk to the dragon, leave the room, kill knights until you have the key, then rescue the dragoon and exit the cave. The dragon will clear the blockage. Go to the screen where you got the final egg.
On this screen go up and right for a heart, then down to continue. When you meet the boss (an easy one, the giant dragon) equip your sword and shield. When he comes close raise your shield. As soon as his breath runs out slash him, then raise your shield again as he comes back. Repeat until he's dead.