The sum of the product and the sum of two positive integers is $454$. Find the largest possible value of the product of their sum and their product.
With word problems, the first step is to translate the words into equations. Let the two numbers be $a$ and $b$. Then their sum is $a+b$ and their product is $ab$. The sum of their product and their sum is $a+b+ab$. So we know  \begin{align*}
ab+a+b&=454\quad\Rightarrow\\
a(b+1)+(b+1)&=454+1\quad\Rightarrow\\
(a+1)(b+1)&=455.
\end{align*}The prime factorization of $455$ is $5\cdot 7\cdot 13$. Since the equation is symmetric with $a$ and $b$, we may (without loss of generality) suppose that $a<b$. Thus $a+1<b+1$, so in each factor pair the smaller factor is equal to $a+1$. We list all possibilities: \begin{tabular}{c|c|c|c}
$a+1$&$b+1$&$a$&$b$\\ \hline
$1$&$455$&$0$&$454$\\
$5$&$91$&$4$&$90$\\
$7$&$65$&$6$&$64$\\
$13$&$35$&$12$&$34$
\end{tabular}We must find the largest possible value of "the product of their sum and their product", or $ab\cdot(a+b)$. We know the first possibility above gives a value of zero, while all the others will be greater than zero. We check: \begin{align*}
4\cdot 90\cdot (4+90)&=4\cdot 90\cdot 94=33840\\
6\cdot 64\cdot (6+64)&=6\cdot 64\cdot 70=26880\\
12\cdot 34\cdot (12+34)&=12\cdot 34\cdot 46=18768.
\end{align*}Thus the largest possible desired value is $\boxed{33840}$, achieved when $(a,b)=(4,90)$.