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

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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:**
$$
-2\left(x + \frac{5}{2}\right)^{2} + \frac{19}{2}
$$
$$ -2\left(x + \frac{5}{2}\right)^{2} + \frac{19}{2} $$

(i) Express in the form .

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