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It looks like you've uploaded a math activity sheet that involves polynomial division using the long division method. Based on the image, the example provided is dividing a cubic polynomial by a linear binomial.

The steps to solve these types of problems are as follows:

1. **Arrange the dividend and divisor** in standard form (descending powers of \(x\)).
2. **Divide the first term of the dividend** by the first term of the divisor, and write that as the first term of the quotient.
3. **Multiply** the divisor by this first term of the quotient, and subtract that product from the dividend.
4. **Bring down** the next term of the dividend, and repeat the process until all terms are divided, or until there is a remainder.

Let's go through one of the exercises together.

### Problem 1:
\[
(4x^3 - x^2 - 6x + 2) \div (x - 1)
\]

**Step 1:** Divide the first term:  
\[
\frac{4x^3}{x} = 4x^2
\]
Now multiply \(4x^2 \times (x - 1) = 4x^3 - 4x^2\).

**Step 2:** Subtract:
\[
(4x^3 - x^2) - (4x^3 - 4x^2) = 3x^2
\]

**Step 3:** Bring down the next term:  
Now we have:
\[
3x^2 - 6x
\]

**Step 4:** Divide:
\[
\frac{3x^2}{x} = 3x
\]
Multiply: \(3x \times (x - 1) = 3x^2 - 3x\).

**Step 5:** Subtract:
\[
(3x^2 - 6x) - (3x^2 - 3x) = -3x
\]

**Step 6:** Bring down the next term:  
Now we have:
\[
-3x + 2
\]

**Step 7:** Divide:
\[
\frac{-3x}{x} = -3
\]
Multiply: \(-3 \times (x - 1) = -3x + 3\).

**Step 8:** Subtract:
\[
(-3x + 2) - (-3x + 3) = -1
\]

So the quotient is \(4x^2 + 3x - 3\) with a remainder of \(-1\).

**Final answer:**
\[
\boxed{4x^2 + 3x - 3 - \frac{1}{x-1}}
\]

Would you like to go through another problem, or get more details on this method?

Here are 5 questions you could explore:
1. How do we check if our long division result is correct?
2. What happens if the divisor is a higher-degree polynomial?
3. Can we apply synthetic division in this case? Why or why not?
4. How does the remainder affect the interpretation of the result?
5. What if the leading coefficient of the divisor was not 1?

**Tip:** Always arrange both the dividend and divisor in decreasing powers of \(x\) before starting the division process for clarity and accuracy.

Use long division to find the quotient and remainder in each of the following polynomial division problems.

Learn how to perform polynomial division using long division with this detailed activity sheet. This example covers dividing a cubic polynomial by a linear binomial and finding both the quotient and remainder. Ideal for high school algebra students (Grades 9-12).

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