Answer
The vertex form of the function is $y=2(x+6)^{2}-47.$ The vertex is $(-6,-47)$.
Work Step by Step
$ y=2x^{2}+24x+25\qquad$ ...factor out $2$ from the first two terms.
$ y=2(x^{2}+12x)+25\qquad$ ...square half the coefficient of $x$.
$(\displaystyle \frac{12}{2})^{2}=6^{2}=36\qquad$ ...complete the square by adding $ 2\cdot 36$ to each side of the expression
$ y+2\cdot 36=2(x^{2}+12x)+25+2(36)\qquad$ ... ...factor out $2$ from the first and third term on the right side of the expression.
$ y+72=2(x+12x+36)^{2}+25\qquad$ ... write $x+12x+36$ as a binomial squared.
$ y+72=2(x+6)+25\qquad$ ...solve for $y$ by adding $-72$ to each side
$ y+72-72=2(x+6)^{2}+25-72\qquad$ ...simplify.
$y=2(x+6)^{2}-47$
The vertex form of a quadratic function is $y=a(x-h)^{2}+k$ where $(h,k)$ is the vertex of the function's graph.
Here, $h=-6,\ k=-47$, so the vertex is $(-6,-47)$
