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Difference of two cubes: `a^3 - b^3`

The formula
a3 - b3 = (a - b)(a2 + ab + b2)
Working forwards - Factorization
For example, the expression x3 - 8y3 can be expanded as follows:
= (x)3 - (2y)3
= (x - 2y)(x2 + (x)(2y) + (2y)2)
= (x - 2y)(x2 + 2xy + 4y2)
.. which is the factored form of x3 - 8y3

Working backwards - Expansion
Since the opposite working of this formula is one step and requires almost no simplification, it is better to see an example of how this formula is applied in a problem like
x2 + 4x + 16
x + 4
In the above example, if we multiply the fraction by  
x - 4
x - 4


x2 + 4x + 16
x + 4
 × 
x  - 4
x - 4
 = 
(x2 + 4x + 16) (x - 4)
(x - 4) (x + 4)



On comparing the numerator with the RHS of the above formula, it is a difference of two cubes, x3 and 43. So, factoring it, you get:
x3 - 43
(x + 4) (x - 4)
.. which is a simplified form of the original expression.