Example 

Multiply 3x⁵ + 4x³ + 2x - 1 and x⁴ + 2x² + 4.

The product is given by

3x⁵ . (x⁴ + 2x² + 4) + 4x³. (x⁴ + 2x² + 4) + 2x .

(x⁴ + 2x² + 4) - 1 . (x⁴ + 2x² + 4)

= 3x⁵ . x⁴ + 3x⁵ . 2x² + 3x⁵ . 4 + 4x³ . x⁴ + 4x³ .

2x² + 4x³ . 4 + 2x . x⁴ + 2x . 2x² + 2x . 4 - x⁴ - 2x² - 4

To simplify the above we employ a rule which we will learn in laws of indices. It states that  x^m . xⁿ = x^m+n

= 3x⁹ + 6x⁷ + 12x⁵ + 4x⁷ + 8x⁵ + 16x³ + 2x⁵ + 4x³ + 8x - x⁴ - 2x² - 4  

Now we collect like terms and simplify them. We obtain 3x⁹ + 10x⁷ + 22x⁵ - x⁴ + 20x³ - 2x² + 8x - 4.