Complete the square on the following quadratic function $y=2 x^{2}+12 x+5$ and determine the vertex and the axis of symmetry.

Step 1 ：Factor out the coefficient of the $x^2$ term: $y=2(x^2+6x)+5$

Step 2 ：Add and subtract the square of half of the coefficient of the $x$ term inside the parentheses: $y=2(x^2+6x+9-9)+5$

Step 3 ：Simplify the expression: $y=2((x+3)^2-9)+5$

Step 4 ：Expand and simplify: $y=2(x+3)^2-18+5$

Step 5 ：Final vertex form: $y=2(x+3)^2-13$

Step 6 ：Find the vertex: $h=-3$, $k=-13$, vertex $\boxed{(-3, -13)}$

Step 7 ：Find the axis of symmetry: $x=\boxed{-3}$