Answer
The maximum product is $81$ and the pair is $\left( -9,-9 \right)$.
Work Step by Step
Let the two numbers be x and y. Then,
$x\ +\ y\ =\ -18$
Now,
$\begin{align}
& x+y=-18 \\
& y=-18-x
\end{align}$
Calculate xy,
$\begin{align}
& xy=x\left( -18-x \right) \\
& xy=-18x-{{x}^{2}} \\
\end{align}$
Which is a downwards opening parabola, which attains its maximum value at $\frac{-b}{2a}$ , where $b=-18$ and $a=-1$.
$\begin{align}
& x=\frac{18}{-2} \\
& x=-9 \\
\end{align}$
Substituting the value of x in the equation (1) we get,
$\begin{align}
& y\ =\ \ -\left( -9 \right)\ \ -\ 18 \\
& =\ \ 9\ \ -\ \ 18 \\
& =\ \ -9
\end{align}$
The maximum product is:
$\begin{align}
& xy\ =\ \left( -9 \right)\ \left( -9 \right) \\
& =\ 81
\end{align}$
The maximum product is $81$ and the pair is $\left( -9,\ -9 \right)$.
