Which of the following is equivalent to `2i(xi-4i^(2))`?

Options

A. `2x-8i`
B. `-2+8i`
C. `-6i`
D. `-8i`

Answer

To solve the expression \(2i(xi - 4i^2)\), we can follow these steps:

### Step 1: Simplify the expression inside the parentheses
We start with the expression \(xi - 4i^2\). We know that \(i^2 = -1\). Therefore, we can substitute \(i^2\) in the expression:

\[ xi - 4i^2 = xi - 4(-1) = xi + 4 \]

### Step 2: Multiply by \(2i\)
Now, we will multiply the simplified expression by \(2i\):

\[ 2i(xi + 4) \]

### Step 3: Distribute \(2i\)
Distributing \(2i\) across the terms inside the parentheses gives us:

\[ 2i \cdot xi + 2i \cdot 4 = 2xi^2 + 8i \]

### Step 4: Substitute \(i^2\) again
Since we know \(i^2 = -1\), we can substitute \(i^2\) in the expression:

\[ 2xi^2 + 8i = 2x(-1) + 8i = -2x + 8i \]

### Final Result
Thus, the expression \(2i(xi - 4i^2)\) simplifies to:

\[ -2x + 8i \]