Write the function $LaTeX: \displaystyle f(x)=x^{2} + 10 x + 31$ in vertex form and identify the vertex.

$LaTeX: \displaystyle h = -\frac{b}{2a}$ and $LaTeX: \displaystyle k=f(h)$ . Using each formula gives $LaTeX: \displaystyle h= -5$ and $LaTeX: \displaystyle k = 6$ . The vertex is located at $LaTeX: \displaystyle (-5,6)$ and the vertex form is $LaTeX: \displaystyle f(x) = a(x-h)^2+k=\left(x + 5\right)^{2} + 6$ .