Answer
$(-4,-6)$
Work Step by Step
To find the vertex, the given function, $
f(x)=x^2+8x+10
,$ should be converted in the form $f(x)=a(x-h)^2+k$.
Grouping the variables together, the given function is equivalent to
\begin{align*}
f(x)&=(x^2+8x)+10
.\end{align*}
Completing the square of the expression with variables by adding $\left(\dfrac{b}{2}\right)^2,$ the expression above is equivalent to
\begin{align*}
f(x)&=\left(x^2+8x+\left(\dfrac{8}{2}\right)^2\right)+\left[10-\left(\dfrac{8}{2}\right)^2\right]
\\\\&=
\left(x^2+8x+\left(4\right)^2\right)+\left[10-\left(4\right)^2\right]
\\&=
\left(x^2+8x+16\right)+\left[10-16\right]
\\&=
\left(x+4\right)^2-6
.\end{align*}Note that $\left(\dfrac{b}{2}\right)^2$ should be subtracted as well to negate the addition of that value when completing the square of the expression with variables.
In the form $f(x)=a(x-h)^2+k$, the equation above is equivalent to
\begin{align*}
f(x)&=\left(x-(-4)\right)^2-6
.\end{align*}
Since the vertex of the quadratic function $f(x)=a(x-h)^2+k$ is given by $(h,k)$, then the vertex of the quadratic function above is $
(-4,-6)
$.
