Let's break down the given expression:

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

Step 1: Multiply the two complex numbers $(3 + 4i)$ and $(3 + 4i)$

We will use the distributive property to expand the product:

$(3 + 4i)(3 + 4i) = 3(3 + 4i) + 4i(3 + 4i)$

Now, distribute each term:

$= 3 \cdot 3 + 3 \cdot 4i + 4i \cdot 3 + 4i \cdot 4i$

This simplifies to:

$= 9 + 12i + 12i + 16i^2$

Since $i^2 = -1$, we substitute that in:

$= 9 + 12i + 12i + 16(-1) = 9 + 24i - 16$

Simplifying:

$= -7 + 24i$

Step 2: Multiply by $(-i)$

Now, multiply $(-i)$ by $(-7 + 24i)$:

$(-i)(-7 + 24i) = (-i)(-7) + (-i)(24i)$

Simplify each term:

$= 7i - 24i^2$

Since $i^2 = -1$, substitute that:

$= 7i - 24(-1) = 7i + 24$

Final Answer:

$24 + 7i$

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Would you like further details on any of the steps or have any other questions? Here are some related questions you might find interesting:

1. How do you multiply two complex numbers in general?
2. What is the concept of the conjugate of a complex number and how is it used?
3. Can you simplify expressions involving powers of $i$?
4. What are some applications of complex numbers in real-world scenarios?
5. How do you divide complex numbers?

Tip: Always remember that $i^2 = -1$ is the fundamental rule when working with complex numbers!