If it has a nontrivial solution, it clearly has one with coprime. But, for example, implies that divides ... can you finish?
No, those two equations are not the same.
What he is saying get the original equation in the form he has it (subtract from both sides. Then you see the RHS you can factor out an x to see that x must divide . But if , then you see that x and y must share at least 1 prime factor, or else x is or y is 0. y=0 yields the trivial solution. has no integer solutions, use the quadratic formula to see why (or rational root test if you want). So they must share a prime factor making them not relatively prime.
Very nice proof bruno.