To divide the polynomial \((4x^4 - 5x^3 + 10x^2 - 3x - 2)\) by \((x + 1)\), we can use polynomial long division. Here’s the process step-by-step:

1. **Divide the first term of the numerator by the first term of the denominator:**  
   \[
   \frac{4x^4}{x} = 4x^3
   \]

2. **Multiply the entire divisor by this quotient term and subtract from the original polynomial:**  
   \[
   (4x^4 - 5x^3 + 10x^2 - 3x - 2) - (4x^3 \cdot (x + 1)) = (4x^4 + 4x^3)
   \]
   Resulting in:  
   \[
   (-9x^3 + 10x^2 - 3x - 2)
   \]

3. **Repeat the process with the new polynomial:**  
   \[
   \frac{-9x^3}{x} = -9x^2
   \]

   Multiply and subtract:  
   \[
   (-9x^3 + 10x^2) - (-9x^2 \cdot (x + 1)) = (-9x^3 - 9x^2)
   \]
   Resulting in:  
   \[
   (19x^2 - 3x - 2)
   \]

4. **Continue the process:**  
   \[
   \frac{19x^2}{x} = 19x
   \]

   Multiply and subtract:  
   \[
   (19x^2 - 3x) - (19x \cdot (x + 1)) = (19x^2 + 19x)
   \]
   Resulting in:  
   \[
   (-22x - 2)
   \]

5. **Final step:**  
   \[
   \frac{-22x}{x} = -22
   \]

   Multiply and subtract:  
   \[
   (-22x - 2) - (-22(x + 1)) = (-22x - 22)
   \]
   Resulting in:  
   \[
   20
   \]

Now, we combine all parts to get:

\[
\text{Quotient: } 4x^3 - 9x^2 + 19x - 22, \quad \text{Remainder: } 20
\]

Thus, the result of the division is:

\[
\frac{4x^4 - 5x^3 + 10x^2 - 3x - 2}{x+1} = 4x^3 - 9x^2 + 19x - 22 + \frac{20}{x+1}
\]

Would you like more details on any step? Or have any specific questions?

### Related Questions:
1. What is polynomial long division, and how does it compare to synthetic division?
2. How do you identify the quotient and remainder in polynomial division?
3. Can polynomial division be used for non-linear divisors?
4. What happens if the remainder is zero in polynomial division?
5. How can polynomial division be applied in calculus?

**Tip:** When dividing polynomials, always arrange terms in descending order of their degree to avoid mistakes in calculations.

Explore the polynomial long division of (4x^4 - 5x^3 + 10x^2 - 3x - 2) by (x + 1) with detailed step-by-step instructions. Understand key algebraic concepts such as dividing polynomials, identifying the quotient and remainder, and applying the long division method. Suitable for Grades 9-12.

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