Common Factors

The distributive properties given in 

(1) a(b + c) = ab + ac     and     (2) a(b - c) = ab - ac

relate the operations of addition and multiplication in (1) and the operations
of subtraction and multiplication in (2).

Note the sum on right side in (1) can be written as the product of a(b + c) and the
difference on the right side of (2) can be written as a(b - c).

When we write a sum or difference as a product we say we FACTOR the sum 
or difference. To factor an expression means we write it as a product.

Consider (1).  Note both terms ab and ac in the sum ab + ac have a factor of a,
that is a is a factor which is common to both terms. Anytime there is a factor
common to every term of an expression you can factor out the common factor
and write the expression as a product. 

Example:   2X + 2Y  = 2(X + Y) 
Example:   3X - 6 = 3X - 3(2) = 3(X - 2) 
Example:  5X² + 3X = 5(X)(X) + 3X = X(5X + 3) 
Example:  30X² Y - 6XY = 6(5)(X)(X)(Y) - 6(1)XY = 6XY (5X - 1) 
Example:  7(X +Y)² + 3(X + Y) = 7(X + Y)(X + Y) + 3(X + Y) = (X + Y)[(7(X + Y) + 3]
= (X + Y)[7X + 7Y + 3]     (Here the common factor is (X + Y).) 

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