Expanding and Simplifying (x + 3i)^2  (2x  3i)^2
This problem involves expanding and simplifying a complex expression. Let's break it down stepbystep:
1. Expanding the Squares
We start by expanding the squares using the formula (a + b)^2 = a^2 + 2ab + b^2 and (a  b)^2 = a^2  2ab + b^2.

Expanding (x + 3i)^2: (x + 3i)^2 = x^2 + 2(x)(3i) + (3i)^2 = x^2 + 6xi + 9i^2

Expanding (2x  3i)^2: (2x  3i)^2 = (2x)^2  2(2x)(3i) + (3i)^2 = 4x^2  12xi + 9i^2
2. Simplifying with i^2 = 1
Remember that the imaginary unit 'i' is defined as the square root of 1, so i^2 = 1. We can substitute this into our expanded expressions:
 (x + 3i)^2: x^2 + 6xi  9
 (2x  3i)^2: 4x^2  12xi  9
3. Combining the Expanded Terms
Now we can subtract the simplified expressions:
(x^2 + 6xi  9)  (4x^2  12xi  9)
Simplify by distributing the negative sign:
x^2 + 6xi  9  4x^2 + 12xi + 9
Combine like terms:
3x^2 + 18xi
4. Final Result
Therefore, the simplified form of (x + 3i)^2  (2x  3i)^2 is 3x^2 + 18xi.
This expression is a complex number, with a real part of 3x^2 and an imaginary part of 18x.