Depending on the graph of a cubic polynomial

First, the solutions on AOPS site depends on the graph of a cubic polynomail. Some calculus is needed to detail the graph of a cubic polynomial.

Second, they are hard to understand, especially on which values should be positive or negative.

I gave a solution without this dependency and it is easy to explain.

I went further to find the polynomial. The polynomial does not satisfy the condition of the problem.

What is wrong with my solution?