Express the following in the form a+ bi
`((4i^(3)-i)^(2))/(2i+1)`

Answer

To express the given expression \(\frac{(4i^3 - i)^2}{2i + 1}\) in the form \(a + bi\), we will follow these steps:

### Step 1: Simplify the numerator
First, we simplify \(4i^3 - i\).

We know that:
\[ i^3 = -i \]
Thus,
\[ 4i^3 = 4(-i) = -4i \]
So,
\[ 4i^3 - i = -4i - i = -5i \]

### Step 2: Square the result
Now, we need to square \(-5i\):
\[ (-5i)^2 = 25i^2 \]
Since \(i^2 = -1\), we have:
\[ 25i^2 = 25(-1) = -25 \]

### Step 3: Write the denominator
The denominator is already given as \(2i + 1\).

### Step 4: Form the complete expression
Now we can write the expression as:
\[ \frac{-25}{2i + 1} \]

### Step 5: Rationalize the denominator
To express this in the form \(a + bi\), we multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{-25(1 - 2i)}{(2i + 1)(1 - 2i)} \]

### Step 6: Calculate the denominator
Now, we calculate the denominator:
\[ (2i + 1)(1 - 2i) = 2i \cdot 1 - 2i \cdot 2i + 1 \cdot 1 - 1 \cdot 2i = 2i - 4i^2 + 1 - 2i \]
Since \(i^2 = -1\), we have:
\[ -4i^2 = 4 \]
Thus, the denominator simplifies to:
\[ 2i - 2i + 1 + 4 = 5 \]

### Step 7: Calculate the numerator
Now, calculate the numerator:
\[ -25(1 - 2i) = -25 + 50i \]

### Step 8: Combine the results
Now we can write the expression:
\[ \frac{-25 + 50i}{5} = -5 + 10i \]

### Final Result
Thus, the expression in the form \(a + bi\) is:
\[ -5 + 10i \]

### Values of A and B
From this, we can identify:
\[ A = -5 \quad \text{and} \quad B = 10 \]