help

To write this out algebraically, we can assign variables to the two numbers. 

 

First number = x, Second number = y

 

The product of the two numbers is x*y or just xy

 

The sum of the two numbers is (x+y) 

 

The sum of the product and the sum of two positive integers can be expressed as: 

 

xy + x + y, and this is equal to 454, xy + x + y = 454 

 

We can solve for all values of x and y, we get: 

 

xy + x + y = 454

 

We add 1 to both sides so we can factor: 

 

xy + x + y + 1 = 455

 

Then we can factor:

 

x(y+1)+y+1 = 455

 

(y+1)(x+1) = 455 = 5 * 7 * 13

 

Their product is the largest only when x = 90 and y = 4 

 

(90*4)(90+4)=33840

 

Other pairs is not as big, for example, when x = 12 and y = 34

 

(34*12)(34+12)=18768

 

To answer your question, the largest possible value is 33840, 

 

I hope this helped,

 

Gavin