1. **State the problem:** Simplify the expression $$5x^2 - 4x - \frac{1}{2}(x-3)(x-1)(x+1)$$. 2. **Recall the formula and rules:** To simplify, first expand the product in the parentheses and then combine like terms. 3. **Expand the product:** \[(x-3)(x-1)(x+1)\] First, multiply \((x-1)(x+1) = x^2 - 1\) (difference of squares). 4. Now multiply \((x-3)(x^2 - 1)\): \[x(x^2 - 1) - 3(x^2 - 1) = x^3 - x - 3x^2 + 3 = x^3 - 3x^2 - x + 3\] 5. **Substitute back:** \[5x^2 - 4x - \frac{1}{2}(x^3 - 3x^2 - x + 3)\] 6. **Distribute the \(-\frac{1}{2}\):** \[5x^2 - 4x - \frac{1}{2}x^3 + \frac{3}{2}x^2 + \frac{1}{2}x - \frac{3}{2}\] 7. **Combine like terms:** - For \(x^3\): \(-\frac{1}{2}x^3\) - For \(x^2\): \(5x^2 + \frac{3}{2}x^2 = \frac{10}{2}x^2 + \frac{3}{2}x^2 = \frac{13}{2}x^2\) - For \(x\): \(-4x + \frac{1}{2}x = -\frac{8}{2}x + \frac{1}{2}x = -\frac{7}{2}x\) - Constant: \(-\frac{3}{2}\) 8. **Final simplified expression:** $$-\frac{1}{2}x^3 + \frac{13}{2}x^2 - \frac{7}{2}x - \frac{3}{2}$$ This is the simplified form of the given expression.