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To solve this division problem using synthetic division, we will use the synthetic division method with \( x = -4 \) (since the divisor is \( x + 4 \)) to divide the polynomial \( x^5 + 5x^4 + 4x^3 + 4x^2 + 15x - 1 \).

### Step-by-Step Solution:

1. **Set up the coefficients of the polynomial:**
   The polynomial \( x^5 + 5x^4 + 4x^3 + 4x^2 + 15x - 1 \) has the coefficients:
   \[ 1, 5, 4, 4, 15, -1 \]

2. **Write the root for synthetic division:**
   Since we are dividing by \( x + 4 \), we use \( -4 \) as the value for synthetic division.

3. **Perform synthetic division:**

   - **Step 1:** Bring down the first coefficient, \( 1 \).
   - **Step 2:** Multiply \( 1 \) by \( -4 \) and add to the next coefficient:
     \[ 5 + (-4) = 1 \]
   - **Step 3:** Multiply \( 1 \) by \( -4 \) and add to the next coefficient:
     \[ 4 + (-4) = 0 \]
   - **Step 4:** Multiply \( 0 \) by \( -4 \) and add to the next coefficient:
     \[ 4 + 0 = 4 \]
   - **Step 5:** Multiply \( 4 \) by \( -4 \) and add to the next coefficient:
     \[ 15 + (-16) = -1 \]
   - **Step 6:** Multiply \( -1 \) by \( -4 \) and add to the last coefficient:
     \[ -1 + 4 = 3 \]

   The synthetic division results are:

   \[
   1 \quad 1 \quad 0 \quad 4 \quad -1 \quad 3
   \]

4. **Interpret the result:**
   The quotient is given by the coefficients \( 1, 1, 0, 4, -1 \), which corresponds to the polynomial:
   \[
   x^4 + x^3 + 0x^2 + 4x - 1
   \]
   And the remainder is \( 3 \).

5. **Write the final answer:**
   \[
   \frac{x^5 + 5x^4 + 4x^3 + 4x^2 + 15x - 1}{x + 4} = x^4 + x^3 + 4x - 1 + \frac{3}{x + 4}
   \]

Let me know if you'd like further details or if you have any questions.

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### Additional Related Questions:

1. How is synthetic division different from long division in polynomials?
2. Why is the root value used in synthetic division the opposite sign of the divisor term?
3. Can synthetic division be used with divisors of higher degrees?
4. How would you interpret a zero remainder in synthetic division?
5. What other methods are there to divide polynomials?

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### Tip:
When using synthetic division, always remember to check that all degrees of the polynomial are present. Insert a zero for any missing terms to maintain correct alignment.

Divide using synthetic division: (x^5 + 5x^4 + 4x^3 + 4x^2 + 15x - 1) / (x + 4)

Learn how to divide the polynomial x^5 + 5x^4 + 4x^3 + 4x^2 + 15x - 1 by x + 4 using synthetic division. This problem involves key algebraic concepts and techniques related to polynomial division, suitable for students in Grades 10-12.

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