Express $$ y=x^{2}+2 x-2 $$ in the form $$ (x+p)^{2}+q $$

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To express the quadratic function $$ y = x^2 + 2x - 2 $$ in the form $$ (x + p)^2 + q $$, we will complete the square.

1. Start with the original equation:
   $$
   y = x^2 + 2x - 2
   $$

2. Focus on the quadratic and linear terms $$ x^2 + 2x $$. To complete the square, we take the coefficient of $$ x $$, which is $$ 2 $$, divide it by $$ 2 $$ to get $$ 1 $$, and then square it to get $$ 1^2 = 1 $$.

3. Rewrite the expression by adding and subtracting this square:
   $$
   y = (x^2 + 2x + 1) - 1 - 2
   $$

4. This simplifies to:
   $$
   y = (x + 1)^2 - 3
   $$

Now, we have expressed $$ y $$ in the desired form:
$$
y = (x + 1)^2 - 3
$$

Thus, we can identify $$ p $$ and $$ q $$:
$$
p = 1, \quad q = -3
$$

The final answer is:
$$
y = (x + 1)^2 - 3
$$
$$ y = (x + 1)^2 - 3 $$

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