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The problem shown in your image asks to divide the polynomial \( x^5 + 5x^4 + 4x^3 + 4x^2 + 14x - 6 \) by \( x + 4 \) using synthetic division.

### Steps for Synthetic Division:
1. **Set up the synthetic division** by using the zero of the divisor. Since the divisor is \( x + 4 \), the zero is \( -4 \).
2. Write the coefficients of the dividend \( x^5 + 5x^4 + 4x^3 + 4x^2 + 14x - 6 \) in order: \( 1, 5, 4, 4, 14, -6 \).
3. Perform synthetic division.

### Synthetic Division Process:
- Write down \( -4 \) (the zero of \( x + 4 \)) on the left.
- List the coefficients \( 1, 5, 4, 4, 14, -6 \) in a row.
- Bring down the first coefficient \( 1 \) as it is.
- Multiply the coefficient by \( -4 \) and add it to the next coefficient:
  - \( 1 \times (-4) = -4 \), add to 5 → \( 5 + (-4) = 1 \).
  - \( 1 \times (-4) = -4 \), add to 4 → \( 4 + (-4) = 0 \).
  - \( 0 \times (-4) = 0 \), add to 4 → \( 4 + 0 = 4 \).
  - \( 4 \times (-4) = -16 \), add to 14 → \( 14 + (-16) = -2 \).
  - \( -2 \times (-4) = 8 \), add to -6 → \( -6 + 8 = 2 \).

The resulting quotient coefficients are \( 1, 1, 0, 4, -2 \), and the remainder is \( 2 \).

### Final Answer:
The quotient is \( x^4 + x^3 + 4x - 2 \), and the remainder is \( 2 \).

Thus, the result of the division is:

\[
x^4 + x^3 + 4x - 2 + \frac{2}{x + 4}
\]

Would you like further details or clarifications on any step?

Here are 5 related questions you could explore:
1. How is synthetic division different from long division for polynomials?
2. What happens if you divide a polynomial by a binomial and the remainder is 0?
3. How can synthetic division be used to find factors of polynomials?
4. Can synthetic division be applied to divisors of degree higher than 1?
5. What is the role of the Remainder Theorem in polynomial division?

**Tip:** Synthetic division works only when the divisor is a linear binomial of the form \( x - c \).

Divide the polynomial x^5 + 5x^4 + 4x^3 + 4x^2 + 14x - 6 by x + 4 using synthetic division.

Explore how to divide the polynomial x^5 + 5x^4 + 4x^3 + 4x^2 + 14x - 6 by x + 4 using synthetic division. This detailed step-by-step solution illustrates key algebra concepts like the remainder theorem and provides quotient and remainder results. Ideal for students in Grades 10-12 and College Algebra courses.

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