4 Express $$ x^{2}+6 x-5 $$ in the form $$ (x+p)^{2}+q $$

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To express the quadratic expression $$ x^{2} + 6x - 5 $$ in the form $$ (x + p)^{2} + q $$, we'll complete the square. Here's how you can do it step-by-step:

1. **Start with the original expression:**
   $$
   x^{2} + 6x - 5
   $$

2. **Focus on the quadratic and linear terms:**
   $$
   x^{2} + 6x
   $$

3. **Complete the square:**
   - Take half of the coefficient of $$ x $$, which is $$ \frac{6}{2} = 3 $$.
   - Square this value: $$ 3^2 = 9 $$.
   - Add and subtract this square inside the expression:
     $$
     x^{2} + 6x + 9 - 9 - 5
     $$

4. **Rewrite the expression:**
   $$
   (x^{2} + 6x + 9) - 14 = (x + 3)^2 - 14
   $$

So, the expression $$ x^{2} + 6x - 5 $$ in the form $$ (x + p)^{2} + q $$ is:

$$
(x + 3)^2 - 14
$$

**Final Answer:**  
After completing the square, the expression is (x + 3)² – 14.
$$ x^{2} + 6x - 5 $$ can be written as $$ (x + 3)^2 - 14 $$.

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