Let's simplify the given expression step by step:

We have:

$2i \left( 4 - i \right)^2$

Step 1: Expand $\left( 4 - i \right)^2$

First, use the formula $(a - b)^2 = a^2 - 2ab + b^2$:

$(4 - i)^2 = 4^2 - 2(4)(i) + (-i)^2$ $= 16 - 8i + i^2$

Since $i^2 = -1$, we get:

$= 16 - 8i - 1$ $= 15 - 8i$

Step 2: Multiply by $2i$

Now, we multiply the result by $2i$:

$2i(15 - 8i) = 2i \cdot 15 - 2i \cdot 8i$ $= 30i - 16i^2$

Since $i^2 = -1$, we get:

$= 30i + 16$

Final Answer:

$16 + 30i$

Do you need further details on any of the steps? Feel free to ask!

Here are some related questions to expand on this topic:

1. What happens when you multiply complex numbers in polar form?
2. How do you simplify expressions with higher powers of $i$?
3. What is the general formula for squaring a binomial involving complex numbers?
4. How do you perform addition and subtraction of complex numbers?
5. What is the significance of $i^2 = -1$ in solving equations?

Tip: When working with powers of $i$, remember that the powers cycle in a pattern: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$, repeating every four terms.