2020 Mathcounts State Prep: Simon's Favorite Factoring Trick

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The most common cases of Simon's Favorite Factoring Trick are:

I:  $xy+x+y+1=(x+1)(y+1)$

II:  $xy-x-y+1=(x-1)(y-1)$

It's easy to learn. Here is the best tutorial online, by none other than Richard Rusczyk.
The method Rusczyk uses at the second half is very nifty. Thanks!!

Questions to ponder:(answer key below)
#1: Both x and y are positive integers and $x>y$. Find all positive integer(s) that $xy+x+y=13$ 
#2: Both x and y are positive integers and $x>y$. Find all positive integer(s) that $2xy+2x-3y=18$
#3: Find the length and the width of a rectangle whose area is equal to its perimeter.
#4: Twice the area of a non-square rectangle equals triple it's perimeter, what is the area of the rectangle? 













Answer key:
#1:  x = 6 and y = 1
#2: ( x, y ) = (4, 2) 
#3: Don't forget square is a kind of rectangle (but not the other way around) so there are two answers: 
4 by 4 and 3 by 6 units. 
#4: One side is 4 units and the other 12 units so the answer is 4 x 12 or 48 square units. 
There is another one, 6 by 6 that would fit if the question doesn't specify non-square rectangle.