Step 1 :\(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)
Step 2 :Here, \(a=1\), \(b=6\), and \(c=7\)
Step 3 :Substituting these values into the quadratic formula, we get:
Step 4 :\(x = \frac{-6 \pm \sqrt{6^{2}-4*1*7}}{2*1}\)
Step 5 :\(x = \frac{-6 \pm \sqrt{36-28}}{2}\)
Step 6 :\(x = \frac{-6 \pm \sqrt{8}}{2}\)
Step 7 :\(x = \frac{-6 \pm 2\sqrt{2}}{2}\)
Step 8 :\(x = -3 \pm \sqrt{2}\)
Step 9 :So, the solutions to the equation \(x^{2}+6 x+7=0\) are \(\boxed{x = -3 + \sqrt{2}}\) and \(\boxed{x = -3 - \sqrt{2}}\)