1. **Stating the problem:** Simplify the expression $$x^{4} - x^{2}y^{2} + 16y^{4}$$. 2. **Formula and rules:** This is a polynomial expression involving powers of $x$ and $y$. We look for factorization patterns such as difference of squares or sum of squares. 3. **Intermediate work:** - The expression is $$x^{4} - x^{2}y^{2} + 16y^{4}$$. - Notice it resembles a quadratic in terms of $x^{2}$: $$ (x^{2})^{2} - (y^{2})(x^{2}) + 16y^{4} $$. 4. **Try to factor as a quadratic:** Let $a = x^{2}$ and $b = y^{2}$, then expression is $$a^{2} - ab + 16b^{2}$$. 5. **Check discriminant:** $$\Delta = (-b)^{2} - 4 \times 1 \times 16b^{2} = b^{2} - 64b^{2} = -63b^{2} < 0$$, so no real factorization over real numbers. 6. **Conclusion:** The expression cannot be factored further over real numbers and is already simplified. **Final answer:** $$x^{4} - x^{2}y^{2} + 16y^{4}$$