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To solve the given polynomial division problems, we will use either long division or factoring where applicable. Let's go through each problem step by step.

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Problem 1: \( (v^2 - 13v + 40) \div (v - 5) \)

#### Step 1: Set up the division
We divide \( v^2 - 13v + 40 \) by \( v - 5 \).

#### Step 2: Perform long division
1. Divide the leading term \( v^2 \) by \( v \): \( \frac{v^2}{v} = v \).
2. Multiply \( v \) by \( v - 5 \): \( v(v - 5) = v^2 - 5v \).
3. Subtract: \( (v^2 - 13v + 40) - (v^2 - 5v) = -8v + 40 \).
4. Divide the leading term \( -8v \) by \( v \): \( \frac{-8v}{v} = -8 \).
5. Multiply \( -8 \) by \( v - 5 \): \( -8(v - 5) = -8v + 40 \).
6. Subtract: \( (-8v + 40) - (-8v + 40) = 0 \).

#### Result:
\[ (v^2 - 13v + 40) \div (v - 5) = v - 8 \]

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Problem 2: \( (8x^2 + 49x + 45) \div (x + 5) \)

#### Step 1: Set up the division
We divide \( 8x^2 + 49x + 45 \) by \( x + 5 \).

#### Step 2: Perform long division
1. Divide the leading term \( 8x^2 \) by \( x \): \( \frac{8x^2}{x} = 8x \).
2. Multiply \( 8x \) by \( x + 5 \): \( 8x(x + 5) = 8x^2 + 40x \).
3. Subtract: \( (8x^2 + 49x + 45) - (8x^2 + 40x) = 9x + 45 \).
4. Divide the leading term \( 9x \) by \( x \): \( \frac{9x}{x} = 9 \).
5. Multiply \( 9 \) by \( x + 5 \): \( 9(x + 5) = 9x + 45 \).
6. Subtract: \( (9x + 45) - (9x + 45) = 0 \).

#### Result:
\[ (8x^2 + 49x + 45) \div (x + 5) = 8x + 9 \]

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Problem 3: \( (5n^2 - 3n - 8) \div (n + 1) \)

#### Step 1: Set up the division
We divide \( 5n^2 - 3n - 8 \) by \( n + 1 \).

#### Step 2: Perform long division
1. Divide the leading term \( 5n^2 \) by \( n \): \( \frac{5n^2}{n} = 5n \).
2. Multiply \( 5n \) by \( n + 1 \): \( 5n(n + 1) = 5n^2 + 5n \).
3. Subtract: \( (5n^2 - 3n - 8) - (5n^2 + 5n) = -8n - 8 \).
4. Divide the leading term \( -8n \) by \( n \): \( \frac{-8n}{n} = -8 \).
5. Multiply \( -8 \) by \( n + 1 \): \( -8(n + 1) = -8n - 8 \).
6. Subtract: \( (-8n - 8) - (-8n - 8) = 0 \).

#### Result:
\[ (5n^2 - 3n - 8) \div (n + 1) = 5n - 8 \]

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Problem 4: \( (a^2 - a - 6) \div (a - 3) \)

#### Step 1: Factor the numerator
The numerator \( a^2 - a - 6 \) can be factored as:
\[ a^2 - a - 6 = (a - 3)(a + 2) \]

#### Step 2: Simplify the division
\[ \frac{a^2 - a - 6}{a - 3} = \frac{(a - 3)(a + 2)}{a - 3} = a + 2 \]

#### Result:
\[ (a^2 - a - 6) \div (a - 3) = a + 2 \]

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Problem 5: \( (k^2 - 7k + 6) \div (k - 6) \)

#### Step 1: Factor the numerator
The numerator \( k^2 - 7k + 6 \) can be factored as:
\[ k^2 - 7k + 6 = (k - 6)(k - 1) \]

#### Step 2: Simplify the division
\[ \frac{k^2 - 7k + 6}{k - 6} = \frac{(k - 6)(k - 1)}{k - 6} = k - 1 \]

#### Result:
\[ (k^2 - 7k + 6) \div (k - 6) = k - 1 \]

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Problem 6: \( (p^2 + 13p + 30) \div (p + 3) \)

#### Step 1: Factor the numerator
The numerator \( p^2 + 13p + 30 \) can be factored as:
\[ p^2 + 13p + 30 = (p + 3)(p + 10) \]

#### Step 2: Simplify the division
\[ \frac{p^2 + 13p + 30}{p + 3} = \frac{(p + 3)(p + 10)}{p + 3} = p + 10 \]

#### Result:
\[ (p^2 + 13p + 30) \div (p + 3) = p + 10 \]

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Problem 7: \( (n^2 - 8n + 16) \div (n - 4) \)

#### Step 1: Recognize the numerator as a perfect square
The numerator \( n^2 - 8n + 16 \) is a perfect square:
\[ n^2 - 8n + 16 = (n - 4)^2 \]

#### Step 2: Simplify the division
\[ \frac{n^2 - 8n + 16}{n - 4} = \frac{(n - 4)^2}{n - 4} = n - 4 \]

#### Result:
\[ (n^2 - 8n + 16) \div (n - 4) = n - 4 \]

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Problem 8: \( (4x^2 - 3x - 7) \div (x + 1) \)

#### Step 1: Set up the division
We divide \( 4x^2 - 3x - 7 \) by \( x + 1 \).

#### Step 2: Perform long division
1. Divide the leading term \( 4x^2 \) by \( x \): \( \frac{4x^2}{x} = 4x \).
2. Multiply \( 4x \) by \( x + 1 \): \( 4x(x + 1) = 4x^2 + 4x \).
3. Subtract: \( (4x^2 - 3x - 7) - (4x^2 + 4x) = -7x - 7 \).
4. Divide the leading term \( -7x \) by \( x \): \( \frac{-7x}{x} = -7 \).
5. Multiply \( -7 \) by \( x + 1 \): \( -7(x + 1) = -7x - 7 \).
6. Subtract: \( (-7x - 7) - (-7x - 7) = 0 \).

#### Result:
\[ (4x^2 - 3x - 7) \div (x + 1) = 4x - 7 \]

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Final Answers

\[
\boxed{
\begin{aligned}
1) & \quad v - 8 \\
2) & \quad 8x + 9 \\
3) & \quad 5n - 8 \\
4) & \quad a + 2 \\
5) & \quad k - 1 \\
6) & \quad p + 10 \\
7) & \quad n - 4 \\
8) & \quad 4x - 7 \\
\end{aligned}
}
\]

I will continue solving the remaining problems if needed. Let me know!