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The problem asks to find the quotient and the remainder when the polynomial:

\[
x^5 + 2x^4 - 15x^3 + 12x + 10
\]

is divided by \( x - 4 \) using synthetic division.

### Steps for Synthetic Division:

1. **Set up the synthetic division**:
   - The divisor \( x - 4 \) has a root of \( 4 \). So, we will use \( 4 \) for the synthetic division.
   - Write down the coefficients of the polynomial: \( 1, 2, -15, 0, 12, 10 \). (Note: The coefficient of \( x^2 \) is \( 0 \) because there is no \( x^2 \) term in the polynomial.)

2. **Perform the synthetic division**:
   - Bring down the first coefficient \( 1 \).
   - Multiply \( 1 \) by \( 4 \), and write the result under the next coefficient.
   - Add down the column.
   - Repeat the process until all coefficients have been processed.

Let's go through the process:

\[
\begin{array}{c|cccccc}
4 & 1 & 2 & -15 & 0 & 12 & 10 \\
  &   & 4 & 24 & 36 & 144 & 624 \\
\hline
  & 1 & 6 & 9 & 36 & 156 & 634 \\
\end{array}
\]

**Explanation of each step**:
- Bring down the \( 1 \) directly.
- Multiply \( 1 \) by \( 4 \) to get \( 4 \), add it to \( 2 \) (result: \( 6 \)).
- Multiply \( 6 \) by \( 4 \) to get \( 24 \), add it to \( -15 \) (result: \( 9 \)).
- Multiply \( 9 \) by \( 4 \) to get \( 36 \), add it to \( 0 \) (result: \( 36 \)).
- Multiply \( 36 \) by \( 4 \) to get \( 144 \), add it to \( 12 \) (result: \( 156 \)).
- Multiply \( 156 \) by \( 4 \) to get \( 624 \), add it to \( 10 \) (result: \( 634 \)).

### Results:
- **Quotient**: \( x^4 + 6x^3 + 9x^2 + 36x + 156 \)
- **Remainder**: \( 634 \)

Thus, the final answer is:
\[
\boxed{\text{Quotient: } x^4 + 6x^3 + 9x^2 + 36x + 156, \; \text{Remainder: } 634}
\]

Would you like more details on the steps or have any questions?

### Related Questions:
1. What is the process of synthetic division when the divisor is \( x + 3 \)?
2. How can you verify the result of a division problem using multiplication?
3. What changes in synthetic division when the divisor root is negative?
4. Can synthetic division be used for divisors other than linear polynomials (like \( x^2 + 1 \))?
5. What is the difference between synthetic division and long division of polynomials?

### Tip:
When performing synthetic division, always double-check the coefficients, especially if any terms are missing in the original polynomial (use 0 for those terms).

Find the quotient and the remainder when x^5 + 2x^4 - 15x^3 + 12x + 10 is divided by x - 4 using synthetic division.

Learn how to find the quotient and remainder when dividing the polynomial x^5 + 2x^4 - 15x^3 + 12x + 10 by x - 4 using synthetic division. This detailed solution breaks down each step, highlighting key algebraic concepts and the division algorithm for polynomials. Ideal for students in Grades 10-12.

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