(i) Express \( -2 x^{2}-10 x-3 \) in the form \( a(x-h)^{2}+k \).

To express the quadratic expression \( -2x^{2} - 10x - 3 \) in the vertex form \( a(x - h)^{2} + k \), follow these steps: 1. **Factor out the coefficient of \( x^2 \) from the first two terms:** \[ -2x^{2} - 10x - 3 = -2(x^{2} + 5x) - 3 \] 2. **Complete the square inside the parentheses:** \[ x^{2} + 5x = \left(x + \frac{5}{2}\right)^{2} - \left(\frac{5}{2}\right)^{2} = \left(x + \frac{5}{2}\right)^{2} - \frac{25}{4} \] 3. **Substitute back into the expression and simplify:** \[ -2\left(\left(x + \frac{5}{2}\right)^{2} - \frac{25}{4}\right) - 3 = -2\left(x + \frac{5}{2}\right)^{2} + \frac{50}{4} - 3 = -2\left(x + \frac{5}{2}\right)^{2} + \frac{25}{2} - 3 \] 4. **Combine the constant terms:** \[ \frac{25}{2} - 3 = \frac{25}{2} - \frac{6}{2} = \frac{19}{2} \] Putting it all together, the expression in vertex form is: \[ -2\left(x + \frac{5}{2}\right)^{2} + \frac{19}{2} \] **Final Answer:** \[ -2\left(x + \frac{5}{2}\right)^{2} + \frac{19}{2} \]