Let's solve the given expression step by step and express it in the form $a + bi$. The expression is:

$(3 - 4xi)(5 + i) - (3 - 4xi)(5 + 4i)$

Step 1: Expand each product

We will use the distributive property to expand both parts.

For the first part:

$(3 - 4xi)(5 + i) = 3(5 + i) - 4xi(5 + i)$ $= 3 \cdot 5 + 3 \cdot i - 4xi \cdot 5 - 4xi \cdot i$ $= 15 + 3i - 20xi - 4x(-1) \quad (\text{since } i^2 = -1)$ $= 15 + 3i - 20xi + 4x$ $= (15 + 4x) + (3 - 20x)i$

For the second part:

$(3 - 4xi)(5 + 4i) = 3(5 + 4i) - 4xi(5 + 4i)$ $= 3 \cdot 5 + 3 \cdot 4i - 4xi \cdot 5 - 4xi \cdot 4i$ $= 15 + 12i - 20xi - 16x(-1) \quad (\text{since } i^2 = -1)$ $= 15 + 12i - 20xi + 16x$ $= (15 + 16x) + (12 - 20x)i$

Step 2: Subtract the second part from the first

Now, subtract the second expression from the first: [ [(15 + 4x) + (3 - 20x)i] - [(15 + 16x) + (12 - 20x)i] ]

Separate the real and imaginary parts:

• Real part: $(15 + 4x) - (15 + 16x) = 15 + 4x - 15 - 16x = -12x$

• Imaginary part: $(3 - 20x) - (12 - 20x) = 3 - 20x - 12 + 20x = -9$

Step 3: Final result

The result in the form $a + bi$ is: $-12x - 9i$

So, the simplified form is: $-12x - 9i$

Would you like more details or explanations on any step? Here are five related questions that can help deepen your understanding:

1. How does the distributive property work in complex number multiplication?
2. Why is $i^2 = -1$ significant in simplifying complex expressions?
3. How can you handle subtraction of complex numbers effectively?
4. What happens when you multiply complex numbers that involve variables like $x$?
5. How can you generalize this process for other complex number expressions?

Tip: Always group real and imaginary parts separately when simplifying complex expressions to avoid errors.