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The image you uploaded contains handwritten math problems related to division of polynomials using synthetic division. Below are the problems and a brief guide on how to solve them:

### Problems:
1. \(\text{Divide } x^3 + 1 \text{ by } x - 1\)
2. \(\text{Divide } 2x^4 + 3x^3 + 2x^2 + 3 \text{ by } x - 1\)
3. \(\text{Divide } 2x^4 - 9x^2 + x - 5 \text{ by } x + 1\)
4. \(\text{Divide } 2x^3 - 3x^2 + x + 1 \text{ by } (x - 1)(x - 1)\)

### Synthetic Division Steps:
1. **Set up the synthetic division**: For a polynomial \( P(x) \) divided by \( x - c \), place \( c \) (the root) to the left and the coefficients of \( P(x) \) to the right.
2. **Bring down the first coefficient**.
3. **Multiply**: Multiply the root by the number just brought down, and place the result under the next coefficient.
4. **Add**: Add the numbers in the column and write the result underneath.
5. **Repeat** until all coefficients are used.
6. The final row gives the coefficients of the quotient polynomial, with the last number being the remainder.

### Example Solution:
Let's solve the first problem as an example:

1. **Problem**: Divide \( x^3 + 1 \) by \( x - 1 \).
   
   - **Step 1**: Set up the division. Here, \( c = 1 \), and the coefficients of the polynomial are \( [1, 0, 0, 1] \).
   - **Step 2**: Perform the synthetic division.

|   | 1 | 0 | 0 | 1 |
|---|---|---|---|---|
| 1 |   | 1 | 1 | 1 |
|   | 1 | 1 | 1 | 2 |

   - **Quotient**: \( x^2 + x + 1 \)
   - **Remainder**: \( 2 \)

This gives us the quotient \( x^2 + x + 1 \) with a remainder of \( 2 \).

Would you like a detailed solution for each problem? If so, let me know!

### Related Questions:
1. How do you handle synthetic division when there are missing degrees in the polynomial?
2. What is the significance of the remainder in synthetic division?
3. How does synthetic division compare to long division of polynomials?
4. Can synthetic division be used for divisors other than \( x - c \)?
5. What happens when the remainder is zero in synthetic division?

### Tip:
Always ensure that the polynomial is in standard form (descending order of powers) and fill in any missing terms with zero coefficients before performing synthetic division.

The query submitted by the user involves division of polynomials using synthetic division. The problems include dividing polynomials by linear divisors such as (x - 1) and (x + 1).

Learn how to divide polynomials using synthetic division with step-by-step solutions. Problems include dividing polynomials such as x^3 + 1 by x - 1 and more. Suitable for high school students.

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