Question 1186857
Here's how to perform synthetic division with complex numbers and rewrite the expression:

**1. Set up the synthetic division:**

Write down the coefficients of the polynomial in the numerator and the value of x that makes the denominator zero (2i in this case):

```
2i |  2   -8i    4    9 - 2i
     |________________________
```

**2. Perform the synthetic division:**

*   Bring down the first coefficient (2):

```
2i |  2   -8i    4    9 - 2i
     |
     ------------------------
       2
```

*   Multiply the value outside (2i) by the number just brought down (2) and write the result under the next coefficient (-8i):

```
2i |  2   -8i    4    9 - 2i
     |      4i
     ------------------------
       2   -4i
```

*   Add the numbers in the second column (-8i and 4i):

```
2i |  2   -8i    4    9 - 2i
     |      4i
     ------------------------
       2   -4i
```

*   Multiply 2i by -4i and write the result under the next coefficient (4):

```
2i |  2   -8i    4    9 - 2i
     |      4i   8
     ------------------------
       2   -4i   12
```

*   Add the numbers in the third column (4 and 8):

```
2i |  2   -8i    4    9 - 2i
     |      4i   8
     ------------------------
       2   -4i   12
```

*   Multiply 2i by 12 and write the result under the last term (9-2i):

```
2i |  2   -8i    4    9 - 2i
     |      4i   8   24i
     ------------------------
       2   -4i   12   9 + 22i
```

*   Add the numbers in the last column:

```
2i |  2   -8i    4    9 - 2i
     |      4i   8   24i
     ------------------------
       2   -4i   12   9 + 22i
```

**3. Interpret the results:**

The numbers in the bottom row (2, -4i, 12) are the coefficients of the quotient, and the last number (9 + 22i) is the remainder. The degree of the quotient is one less than the degree of the original polynomial.

Therefore, the quotient q(x) is 2x² - 4ix + 12, and the remainder r(x) is 9 + 22i. The divisor d(x) is x - 2i.

**4. Write the expression:**

The expression can be written as:

2x² - 4ix + 12 + (9 + 22i) / (x - 2i)