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{array}{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{array} 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) $.