1. **State the problem:** Find the difference between the two expressions: $$\left(\frac{3}{4}x^{2} + 5x - 2\right) - \left(\frac{1}{2} x^{2} - \frac{3}{4} x + \frac{7}{2}\right)$$ 2. **Apply the subtraction:** Distribute the minus sign to the second polynomial: $$\frac{3}{4}x^{2} + 5x - 2 - \frac{1}{2} x^{2} + \frac{3}{4} x - \frac{7}{2}$$ 3. **Group like terms:** $$\left(\frac{3}{4}x^{2} - \frac{1}{2} x^{2}\right) + \left(5x + \frac{3}{4} x\right) + \left(-2 - \frac{7}{2}\right)$$ 4. **Simplify each group:** - For $x^{2}$ terms: $$\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}$$ - For $x$ terms: $$5 + \frac{3}{4} = \frac{20}{4} + \frac{3}{4} = \frac{23}{4}$$ - For constants: $$-2 - \frac{7}{2} = -\frac{4}{2} - \frac{7}{2} = -\frac{11}{2}$$ 5. **Write the final simplified expression:** $$\frac{1}{4} x^{2} + \frac{23}{4} x - \frac{11}{2}$$ 6. **Match with the options:** This corresponds to option C. **Final answer:** C $\frac{1}{4}x^{2} + \frac{23}{4} x - \frac{11}{2}$