Show your work: use the distributive property in two different ways to find the product of 127 and 32.

The distributive property: a(b + c) = ab + ac

127 × 32 = 127 × (30 + 2) = 127 × 30 + 127 × 2 = (130 - 3)  × 30 + (130 - 3) × 2

=(130)(30) - (3)(30) + (130)(2) - (3)(2) = 3900 - 90 + 260 - 6

= 3810 + 254 = 4064

127 × 32 = (130 - 3) × 32 = 130 × 32 - 3 × 32 = 130 × (30 + 2) - 96

= (130)(30) + (130)(2) - 96 = 3900 + 260 - (100 - 4) = 4160 - 100 + 4

= 4060 + 4 = 4064