We need to perform polynomial long division for three problems: a) $(x^3 - 2x + 1) \div (x - 4)$ b) $(x^3 + 2x^2 - 6x + 1) \div (x + 2)$ c) $(2x^3 + 5x^2 - 4x - 5) \div (2x + 1)$

1. Problem Description

We need to perform polynomial long division for three problems:

a) $(x^3 - 2x + 1) \div (x - 4)$

b) $(x^3 + 2x^2 - 6x + 1) \div (x + 2)$

c) $(2x^3 + 5x^2 - 4x - 5) \div (2x + 1)$

2. Solution Steps

a) $(x^3 - 2x + 1) \div (x - 4)$

First, rewrite the dividend with missing terms: $x^3 + 0x^2 - 2x + 1$.

Now perform long division:

```

x^2 + 4x + 14

x - 4 | x^3 + 0x^2 - 2x + 1

-(x^3 - 4x^2)

----------------

4x^2 - 2x

-(4x^2 - 16x)

----------------

14x + 1

-(14x - 56)

----------------

57

```

Therefore, $(x^3 - 2x + 1) \div (x - 4) = x^2 + 4x + 14 + \frac{57}{x-4}$

b) $(x^3 + 2x^2 - 6x + 1) \div (x + 2)$

Now perform long division:

```

x^2 - 6

x + 2 | x^3 + 2x^2 - 6x + 1

-(x^3 + 2x^2)

----------------

0x^2 - 6x

-(-6x - 12)

----------------

13

```

Therefore, $(x^3 + 2x^2 - 6x + 1) \div (x + 2) = x^2 - 6 + \frac{13}{x+2}$

c) $(2x^3 + 5x^2 - 4x - 5) \div (2x + 1)$

Now perform long division:

```

x^2 + 2x - 3

2x + 1 | 2x^3 + 5x^2 - 4x - 5

-(2x^3 + x^2)

----------------

4x^2 - 4x

-(4x^2 + 2x)

----------------

-6x - 5

-(-6x - 3)

----------------

-2

```

Therefore, $(2x^3 + 5x^2 - 4x - 5) \div (2x + 1) = x^2 + 2x - 3 - \frac{2}{2x+1}$

3. Final Answer

a) $x^2 + 4x + 14 + \frac{57}{x-4}$

b) $x^2 - 6 + \frac{13}{x+2}$

c) $x^2 + 2x - 3 - \frac{2}{2x+1}$