$x^4-2x^3+x=y^4+3y^2+y$ in the set of integers

Question

The task is to solve the equation $x^4-2x^3+x=y^4+3y^2+y$ in integers.

I expect is has something to do with factorizing but have no concrete idea; any help? thx guys

Answer 1

It is a problem about inequalities, more than factorization.

Step 1) Prove that there exists a natural number $k$ such that $|x-y|>k$ implies that there are no solutions to the given equation;

Step 2) For any $j$ in the range $[-k,k]$, find the integer solutions to the cubic equation given by setting $y=x+j$;

Step 3) Profit. The only solutions are $(x,y)\in\{(0,0),(1,0),(3,2),(-2,2)\}$.