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To solve this problem using synthetic division, we are dividing \( x^4 - 4x^3 + 2x + 3 \) by \( x - 1 \). Let's go through the synthetic division step-by-step.

### Step 1: Set up the synthetic division
Since we’re dividing by \( x - 1 \), we use \( 1 \) as the synthetic divisor.

Write down the coefficients of \( x^4 - 4x^3 + 0x^2 + 2x + 3 \). Note that there is no \( x^2 \) term, so we use \( 0 \) as the placeholder:

\[
1, -4, 0, 2, 3
\]

### Step 2: Perform synthetic division
1. Bring down the first coefficient, \( 1 \).
2. Multiply \( 1 \) by \( 1 \) (the divisor) and add to the next coefficient:
   - \( 1 \times 1 = 1 \)
   - \(-4 + 1 = -3\)
3. Multiply \(-3\) by \( 1 \) and add to the next coefficient:
   - \(-3 \times 1 = -3\)
   - \(0 + (-3) = -3\)
4. Multiply \(-3\) by \( 1 \) and add to the next coefficient:
   - \(-3 \times 1 = -3\)
   - \(2 + (-3) = -1\)
5. Multiply \(-1\) by \( 1 \) and add to the last coefficient:
   - \(-1 \times 1 = -1\)
   - \(3 + (-1) = 2\)

After these steps, we obtain the following results:

\[
1, -3, -3, -1, 2
\]

### Step 3: Interpret the results
The synthetic division results give us the coefficients of the quotient and the remainder:
- The quotient is \( x^3 - 3x^2 - 3x - 1 \).
- The remainder is \( 2 \).

### Final Answer
- **Quotient**: \( x^3 - 3x^2 - 3x - 1 \)
- **Remainder**: \( 2 \)

Would you like further explanation on synthetic division or have additional questions?

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#### Follow-up Questions:
1. How does synthetic division differ from long division for polynomials?
2. Can synthetic division be used if the divisor is not in the form \( x - c \)?
3. What are some practical applications of synthetic division?
4. How would the answer change if the divisor was \( x + 1 \) instead?
5. How can we check if our quotient and remainder are correct?

#### Tip:
Remember to always include zero placeholders for missing terms in polynomials when using synthetic division.

Find the quotient and remainder using synthetic division for the polynomial (x^4 - 4x^3 + 2x + 3) divided by (x - 1).

Learn how to find the quotient and remainder of the polynomial x^4 - 4x^3 + 2x + 3 divided by x - 1 using synthetic division. This step-by-step solution covers synthetic division, polynomial division, and the remainder theorem, making it suitable for high school students.

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