# Simon's Favorite Factoring Trick

## Introduction

Simon's Favorite Factoring Trick (abbreviated SFFT) is a special factorization first popularized by AoPS user Simon Rubenstein-Salzedo. This appears to be the thread where Simon's favorite factoring trick was first introduced.

## Statement of the factorization

The general statement of SFFT is: ${xy}+{xk}+{yj}+{jk}=(x+j)(y+k)$. More oftenly however SFFT is introduced as $xy + x + y + 1 = (x+1)(y+1)$ or $xy - x - y +1 = (x-1)(y-1)$.

## Applications

This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually ${x}$ and ${y}$ are variables and $j,k$ are known constants. Also it is typically necessary to add the ${j}{k}$ term to both sides to perform the factorization.

## Examples

(AIME 1987/5) $m$ and $n$ are integers such that $m^2 + 3m^2n^2 = 30n^2 + 517$. Find $3m^2n^2$.

Outline Solution: Rearrange to $m^2 + 3m^2n^2 -30n^2= 517$. The key step is changing the equation to $m^2 + 3m^2n^2 -30n^2-10= 507$, where the equation factors to $(3n^2 + 1)(m^2 - 10) = 507 = 3\cdot 13^2$. This makes things much simpler. The rest of the problem is left as an exercise to the reader.