Answer
The vertex form of the function is $y=(x-(-7))^{2}+(-4).$ The vertex is $(-7,-4)$.
Work Step by Step
$ y=x^{2}+14x+45\qquad$ ...write in form of $x^{2}+bx=c$ (add $-45$ to each side).
$ y-45=x^{2}+14x\qquad$ ...square half the coefficient of $x$.
$(\displaystyle \frac{14}{2})^{2}=(7)^{2}=49\qquad$ ...complete the square by adding $49$ to each side of the expression
$ y-45+49=x^{2}+14x+49\qquad$ ... write $x^{2}+14x+49$ as a binomial squared.
$ y+4=(x+7)^{2}\qquad$ ...add $-4$ to each side of the expression
$ y=(x+7)^{2}-4\qquad$ ...write in vertex form $y=a(x-h)^{2}+k$.
$y=(x-(-7))^{2}+(-4)$
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, $h=-7,\ k=-4$, so the vertex is $(-7,-4)$
