Answer
$c(p) = \frac{2}{7}\left(p-\frac{35}{4} \right)^2-\frac{1249}{56}$
Work Step by Step
The conversion of the function to the standard vertex form is self explanatory. All mathematical steps are shown below. $$
\begin{aligned}
c(p)&=\frac{2}{7} p^2-5 p-\frac{3}{7}\\
& = \frac{2}{7}\left( p^2-\frac{7}{2}\cdot 5p \right) -\frac{3}{7}\\
&= \frac{2}{7}\left( p^2-\frac{35}{2}\right) -\frac{3}{7}\\
& = \frac{2}{7}\left[ p^2-\frac{35}{2}p+\left( \frac{35}{4}\right)^2\right]-\frac{3}{7}-\frac{2}{7}\cdot \left( \frac{35}{4}\right)^2\\
& = \frac{2}{7}\left(p-\frac{35}{4} \right)^2-\frac{3}{7}-\frac{2\cdot 1225}{7\cdot 16} \\
& = \frac{2}{7}\left(p-\frac{35}{4} \right)^2-\frac{3}{7}\cdot\frac{8}{8}-\frac{175}{ 8}\cdot \frac{7}{7}\\
& = \frac{2}{7}\left(p-\frac{35}{4} \right)^2-\frac{1249}{56}.
\end{aligned}
$$
