Answer:

Sure, let's perform long division and synthetic division to divide \( P(x) = 2x^4 - 7x^3 - 2x^2 + 6x + 3 \) by \( D(x) = x - 2 \).

**Long Division:**

PICTURE BELOW

So, \( P(x) = (2x³ - 3x² - 8x + 15) \) with a remainder of -41.

**Synthetic Division:**

Since \( D(x) = x - 2 \), we use 2 as the divisor.

1. Bring down the coefficient of the highest power term: `2`.

2. Multiply the divisor by the result and write it below the next term: \(2 \times (x - 2) = 2x - 4\).

3. Subtract this new value from the next column: \((-7x³ - 2x² + 6x) - (2x - 4) = -7x³ - 2x² + 6x - 2x + 4\).

4. Repeat the process: \(2x³ - 3x² - 8x + 15\) divided by \(x - 2\) is \(2x² - 3x - 8\) with a remainder of -41.

Both methods yield the same result: \(P(x) = (x - 2)(2x³ - 3x² - 8x + 15) - 41\).