To solve this problem using synthetic division, follow these steps:

We are dividing:

$4x^5 + x^4 - 4x^3 + x^2 - 3x - 1 \quad \text{by} \quad (x + 2)$

Step 1: Set up synthetic division.

• First, identify the root of the divisor. Since the divisor is $x + 2$, the root is $-2$.
• Now, write down the coefficients of the polynomial: $4, 1, -4, 1, -3, -1$.

Step 2: Perform the synthetic division process.

1. Bring down the first coefficient $4$.
2. Multiply $4$ by $-2$ (the root), and write it under the next coefficient: $4 \times -2 = -8$.
3. Add this to the next coefficient: $1 + (-8) = -7$.
4. Multiply $-7$ by $-2$: $-7 \times -2 = 14$.
5. Add this to the next coefficient: $-4 + 14 = 10$.
6. Multiply $10$ by $-2$: $10 \times -2 = -20$.
7. Add this to the next coefficient: $1 + (-20) = -19$.
8. Multiply $-19$ by $-2$: $-19 \times -2 = 38$.
9. Add this to the next coefficient: $-3 + 38 = 35$.
10. Multiply $35$ by $-2$: $35 \times -2 = -70$.
11. Add this to the next coefficient: $-1 + (-70) = -71$.

Step 3: Write the quotient and remainder.

The quotient is given by the numbers on the top row: $4, -7, 10, -19, 35$, representing:

$4x^4 - 7x^3 + 10x^2 - 19x + 35$

The remainder is $-71$, so the final result is:

$4x^4 - 7x^3 + 10x^2 - 19x + 35 - \frac{71}{x+2}$

This is the simplified answer.

Would you like more details on any of the steps?

Here are five related questions to expand your understanding:

1. How does synthetic division differ from long division of polynomials?
2. What is the significance of the remainder in synthetic division?
3. How would the process change if the divisor was $x - 2$ instead of $x + 2$?
4. Can synthetic division be used for divisors of higher degree, like $x^2 + 2x + 3$?
5. How does synthetic division apply when the leading coefficient of the divisor is not 1?

Tip: Synthetic division works only when the divisor is a linear polynomial of the form $x - c$.