Let's go through each part of the problems provided:

Problem I: Completing the Square

For each quadratic equation, we'll complete the square to solve for $x$:

1. (a) $x^2 + 6x + 3 = 0$:

   1. Move the constant term: $x^2 + 6x = -3$.
   2. Complete the square: $\left(\frac{6}{2}\right)^2 = 9$.
   3. Add 9 to both sides: $x^2 + 6x + 9 = 6$.
   4. Rewrite as a square: $(x + 3)^2 = 6$.
   5. Solve for $x$: $x = -3 \pm \sqrt{6}$.

2. (b) $s^2 + 2s - 6 = 0$:

   1. Move the constant term: $s^2 + 2s = 6$.
   2. Complete the square: $\left(\frac{2}{2}\right)^2 = 1$.
   3. Add 1 to both sides: $s^2 + 2s + 1 = 7$.
   4. Rewrite as a square: $(s + 1)^2 = 7$.
   5. Solve for $s$: $s = -1 \pm \sqrt{7}$.

3. (c) $x^2 + 4x - 2 = 0$:

   1. Move the constant term: $x^2 + 4x = 2$.
   2. Complete the square: $\left(\frac{4}{2}\right)^2 = 4$.
   3. Add 4 to both sides: $x^2 + 4x + 4 = 6$.
   4. Rewrite as a square: $(x + 2)^2 = 6$.
   5. Solve for $x$: $x = -2 \pm \sqrt{6}$.

4. (d) $t^2 - 8t - 5 = 0$:

   1. Move the constant term: $t^2 - 8t = 5$.
   2. Complete the square: $\left(\frac{-8}{2}\right)^2 = 16$.
   3. Add 16 to both sides: $t^2 - 8t + 16 = 21$.
   4. Rewrite as a square: $(t - 4)^2 = 21$.
   5. Solve for $t$: $t = 4 \pm \sqrt{21}$.

5. (e) $2(z + 9) = 1$:

   1. Distribute and rearrange: ( 2z + 18 = 1 \Rightarrow z^2 +