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Let's solve part **2(b)** of the question using polynomial long division.

### 2(b)(i): 
The division is:

\[
\frac{x^4 + 7x^3 + 13x^2 + 2x - 2}{x^2 + 3x - 1}
\]

We will divide the polynomials step-by-step:

#### Step 1:
- Divide the leading term of the numerator \(x^4\) by the leading term of the denominator \(x^2\), giving \(x^2\).

#### Step 2:
- Multiply \(x^2\) by \(x^2 + 3x - 1\), resulting in \(x^4 + 3x^3 - x^2\).
- Subtract this from the original polynomial: 
  \[
  (x^4 + 7x^3 + 13x^2 + 2x - 2) - (x^4 + 3x^3 - x^2) = 4x^3 + 14x^2 + 2x - 2
  \]

#### Step 3:
- Divide the leading term \(4x^3\) by \(x^2\), giving \(4x\).
- Multiply \(4x\) by \(x^2 + 3x - 1\), giving \(4x^3 + 12x^2 - 4x\).
- Subtract this from the result: 
  \[
  (4x^3 + 14x^2 + 2x - 2) - (4x^3 + 12x^2 - 4x) = 2x^2 + 6x - 2
  \]

#### Step 4:
- Divide the leading term \(2x^2\) by \(x^2\), giving \(+2\).
- Multiply \(2\) by \(x^2 + 3x - 1\), giving \(2x^2 + 6x - 2\).
- Subtract this:
  \[
  (2x^2 + 6x - 2) - (2x^2 + 6x - 2) = 0
  \]

Thus, the quotient is:

\[
x^2 + 4x + 2
\]

### 2(b)(ii):
The division is:

\[
\frac{x^4 + 3x^3 - 32x^2 - 17x + 3}{x^2 - 4x - 3}
\]

#### Step 1:
- Divide the leading term \(x^4\) by \(x^2\), giving \(x^2\).
- Multiply \(x^2\) by \(x^2 - 4x - 3\), giving \(x^4 - 4x^3 - 3x^2\).
- Subtract this:
  \[
  (x^4 + 3x^3 - 32x^2 - 17x + 3) - (x^4 - 4x^3 - 3x^2) = 7x^3 - 29x^2 - 17x + 3
  \]

#### Step 2:
- Divide the leading term \(7x^3\) by \(x^2\), giving \(7x\).
- Multiply \(7x\) by \(x^2 - 4x - 3\), giving \(7x^3 - 28x^2 - 21x\).
- Subtract:
  \[
  (7x^3 - 29x^2 - 17x + 3) - (7x^3 - 28x^2 - 21x) = -x^2 + 4x + 3
  \]

#### Step 3:
- Divide the leading term \(-x^2\) by \(x^2\), giving \(-1\).
- Multiply \(-1\) by \(x^2 - 4x - 3\), giving \(-x^2 + 4x + 3\).
- Subtract:
  \[
  (-x^2 + 4x + 3) - (-x^2 + 4x + 3) = 0
  \]

Thus, the quotient is:

\[
x^2 + 7x - 1
\]

Would you like any further details or have any questions? Here are some related questions to explore:

1. What is the general process of polynomial long division?
2. How can you verify your result after performing polynomial division?
3. What is synthetic division and how does it compare to polynomial long division?
4. How would these solutions change if you used a different method like factoring?
5. How does the degree of the numerator and denominator affect the division?

**Tip:** Always check the degree of both the numerator and the denominator before performing polynomial division to understand the expected degree of the quotient.

Use polynomial division to simplify the expression: (x^4 + 3x^3 - 32x^2 - 17x + 3) / (x^2 - 4x - 3).

Learn how to simplify the polynomial expression (x^4 + 3x^3 - 32x^2 - 17x + 3) using polynomial long division. This detailed step-by-step solution covers key algebraic concepts such as polynomial division and the division algorithm for polynomials. Suitable for students in Grades 10-12.

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