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To solve this problem using synthetic division, we will divide \( x^3 + 4x^2 + 8x + 5 \) by \( x + 1 \).

### Steps for Synthetic Division:

1. **Set up the synthetic division**:
   - The divisor is \( x + 1 \), which has a root of \( x = -1 \).
   - Write down the coefficients of the polynomial \( x^3 + 4x^2 + 8x + 5 \): these are \( 1, 4, 8, \) and \( 5 \).

2. **Perform synthetic division**:
   - Write \(-1\) to the left and the coefficients \( 1, 4, 8, 5 \) in a row.
   - Bring down the first coefficient \(1\) as it is.
   - Multiply \(1\) by \(-1\) and write the result under the next coefficient. Add \(4 + (-1) = 3\).
   - Multiply \(3\) by \(-1\) to get \(-3\), and add to the next coefficient \(8 + (-3) = 5\).
   - Multiply \(5\) by \(-1\) to get \(-5\), and add to the next coefficient \(5 + (-5) = 0\).

Here’s how it looks in steps:

\[
\begin{array}{c|cccc}
-1 & 1 & 4 & 8 & 5 \\
   &   & -1 & -3 & -5 \\
\hline
   & 1 & 3 & 5 & 0 \\
\end{array}
\]

3. **Interpret the result**:
   - The bottom row, \(1, 3, 5, 0\), represents the coefficients of the quotient and remainder.
   - The quotient is \( x^2 + 3x + 5 \), and the remainder is \( 0 \).

### Final Answer:
\[
\frac{x^3 + 4x^2 + 8x + 5}{x + 1} = x^2 + 3x + 5
\]

There is no remainder, so the division is exact.

Would you like more details on any step or have any questions?

Here are some related questions you might find interesting:
1. How does synthetic division differ from long division for polynomials?
2. Can synthetic division be used for any divisor, or only linear ones?
3. What does it mean if there's a remainder in synthetic division?
4. How can synthetic division help in finding polynomial roots?
5. How do we handle synthetic division if the polynomial has missing terms?

**Tip:** Synthetic division is most useful when dividing by linear divisors of the form \(x - c\).

Use synthetic division to divide x^3 + 4x^2 + 8x + 5 by x + 1.

Learn how to use synthetic division to divide the polynomial x^3 + 4x^2 + 8x + 5 by x + 1. This problem covers important algebraic concepts, including polynomial division and the division algorithm. Suitable for students in Grades 9-11.

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