1. **State the problem:** Find the sum of the polynomials $$\frac{1}{2}x^2 - 3$$ and $$2x^2 - \frac{1}{4}x + \frac{5}{2}$$. 2. **Write the expression:** $$\left(\frac{1}{2}x^2 - 3\right) + \left(2x^2 - \frac{1}{4}x + \frac{5}{2}\right)$$ 3. **Combine like terms:** - Combine the $$x^2$$ terms: $$\frac{1}{2}x^2 + 2x^2 = \frac{1}{2}x^2 + \frac{4}{2}x^2 = \frac{5}{2}x^2$$ - Combine the $$x$$ terms: $$0 - \frac{1}{4}x = -\frac{1}{4}x$$ - Combine the constants: $$-3 + \frac{5}{2} = -\frac{6}{2} + \frac{5}{2} = -\frac{1}{2}$$ 4. **Write the final simplified polynomial:** $$\frac{5}{2}x^2 - \frac{1}{4}x - \frac{1}{2}$$ 5. **Answer:** The sum of the polynomials is $$\frac{5}{2}x^2 - \frac{1}{4}x - \frac{1}{2}$$, which corresponds to option C.