Answer
$s(p) = (p+7)^2-14 $
Work Step by Step
The conversion of the function to the standard vertex form $s(p)=a(p-h)^2+k$ is self explanatory. All mathematical steps are shown below. $$
\begin{aligned}
s(p)&=p^2+14 p+35\\
& = p^2+14 p t+\left(\frac{14}{2}\right)^2 +35-\left(\frac{14}{2}\right)^2 \\
& =\left(p^2+14 p+7^2\right)+35-7^2 \\
& =(p+7)^2-14.
\end{aligned}
$$ The vertex form of the given function is $$s(p)=(p+7)^2-14.$$
