If you want to prove an inequality (or an equality), generally the best way to write it is as a single chain of inequalities and equalities that bridges the original two quantities.

For example, here you can write:

For all x > 2,

|2x² + x ln x + 4| = 2x² + x ln x + 4
                           ≤ 2x² + x⋅x + 4
                           = 3x² + 4
                           < 3x² + x²
                           = 4x²
                           = 4|x²|.

This chain of equalities and inequalities shows that |2x² + x ln x + 4| < 4|x²| for all x > 2.

Does that make sense? Do you see why this chain proves that inequality?