#### Answer

The expression for the product of two numbers $P$ which sums $50$ as a function of one of the numbers, $x$ is $P\left( x \right)=-{{x}^{2}}+50x$.

#### Work Step by Step

Let the first number be x and the second number be $y$.
The sum of the number x and y is $50$.
$x+y=50$.
Rearrange for y.
$y=50-x$.
Consider the product of two numbers.
$P=xy$
Substitute $50-x$ for $y$.
$P=x\left( 50-x \right)$
The product is a function of x which can also be expressed as,
$\begin{align}
& P\left( x \right)=x\left( 50-x \right) \\
& =-{{x}^{2}}+50x
\end{align}$
Hence, the expression for the product of two numbers $P$ which sums to $50$ as a function of one of the numbers, $x$ is $P\left( x \right)=-{{x}^{2}}+50x$.