1. **State the problem:** Simplify or analyze the expression $$\frac{5}{2} x^6 - \frac{1}{2} x^4$$. 2. **Identify the terms:** The expression has two terms: $$\frac{5}{2} x^6$$ and $$- \frac{1}{2} x^4$$. 3. **Factor common terms:** Both terms share a common factor of $$\frac{1}{2} x^4$$. 4. **Factor out the common factor:** $$\frac{5}{2} x^6 - \frac{1}{2} x^4 = \frac{1}{2} x^4 \left( \cancel{5} x^{6-4} - \cancel{1} \right) = \frac{1}{2} x^4 (5 x^2 - 1)$$ 5. **Final simplified form:** $$\frac{1}{2} x^4 (5 x^2 - 1)$$ This is the factored form of the original expression, which is often easier to work with for further operations like solving or graphing.