Simplifying Complex Expressions
This article will walk through the steps to simplify the complex expression: (8 + 5i)(3 + 2i)  (4 + i)(4  i).
Understanding Complex Numbers
Before we begin, let's understand the basics of complex numbers:
 Complex numbers are numbers of the form a + bi, where:
 a is the real part
 b is the imaginary part
 i is the imaginary unit, defined as the square root of 1 (i² = 1)
Simplifying the Expression

Expand the products:
 (8 + 5i)(3 + 2i) = 24 + 16i + 15i + 10i²
 (4 + i)(4  i) = 16  i²

Substitute i² with 1:
 24 + 16i + 15i + 10(1) = 14 + 31i
 16  (1) = 17

Subtract the simplified terms:
 (14 + 31i)  17 = 3 + 31i
Final Answer
Therefore, the simplified form of the expression (8 + 5i)(3 + 2i)  (4 + i)(4  i) is 3 + 31i.