Answer
The vertex form of the function is $g(x)=(x+\displaystyle \frac{7}{2})^{2}-\frac{41}{4}.$ The vertex is $(\displaystyle \frac{7}{2},-\displaystyle \frac{41}{4})$.
Work Step by Step
$ g(x)=x^{2}+7x+2\qquad$ ...first, prepare to complete the square.
$ g(x)+?=(x^{2}+7x+?)+2\qquad$ ...square half the coefficient of $x$.
$(\displaystyle \frac{7}{2})^{2}=\frac{49}{4}\qquad$ ...complete the square by adding $\displaystyle \frac{49}{4}$ to each side of the expression
$ g(x)+\displaystyle \frac{49}{4}=x^{2}+7x+\frac{49}{4}+2\qquad$ ... write $x^{2}+7x+\displaystyle \frac{49}{4}$ as a binomial squared.
$ g(x)+\displaystyle \frac{49}{4}=(x+\frac{7}{2})^{2}+2\qquad$ ...add $-\displaystyle \frac{49}{4}$ to each side of the expression
$ g(x)+\displaystyle \frac{49}{4}-\frac{49}{4}=(x+\frac{7}{2})^{2}+2-\frac{49}{4}\qquad$ ...simplify.
$g(x)=(x+\displaystyle \frac{7}{2})^{2}-\frac{41}{4}$
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, $h=\displaystyle \frac{7}{2},\ k=-\displaystyle \frac{41}{4}$, so the vertex is $(\displaystyle \frac{7}{2},-\displaystyle \frac{41}{4})$
