Algebraic expressions with polynomial operations

Problem Analysis:

The task involves simplifying three polynomial expressions. Each expression requires combining like terms after performing the indicated operations (addition or subtraction). Let's solve each part step by step.

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Part 1: Simplify \( (4x^2 + 6x^3 + x - 7) - (2x^2 + 6x + 3) \)

#### Step 1: Distribute the negative sign
Distribute the negative sign across the second polynomial:
\[
(4x^2 + 6x^3 + x - 7) - (2x^2 + 6x + 3) = 4x^2 + 6x^3 + x - 7 - 2x^2 - 6x - 3
\]

#### Step 2: Combine like terms
Group the terms by their powers of \( x \):
- \( x^3 \)-terms: \( 6x^3 \)
- \( x^2 \)-terms: \( 4x^2 - 2x^2 \)
- \( x \)-terms: \( x - 6x \)
- Constant terms: \( -7 - 3 \)

Simplify each group:
- \( x^3 \)-terms: \( 6x^3 \)
- \( x^2 \)-terms: \( 4x^2 - 2x^2 = 2x^2 \)
- \( x \)-terms: \( x - 6x = -5x \)
- Constant terms: \( -7 - 3 = -10 \)

Combine all simplified terms:
\[
6x^3 + 2x^2 - 5x - 10
\]

#### Final Answer for Part 1:
\[
\boxed{6x^3 + 2x^2 - 5x - 10}
\]

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Part 2: Simplify \( (3x^2 + 9x + x^5 + 6) + (2x^5 + x^3 - 2x^2 + x - 3) \)

#### Step 1: Remove parentheses
Since this is an addition, we can simply remove the parentheses:
\[
3x^2 + 9x + x^5 + 6 + 2x^5 + x^3 - 2x^2 + x - 3
\]

#### Step 2: Combine like terms
Group the terms by their powers of \( x \):
- \( x^5 \)-terms: \( x^5 + 2x^5 \)
- \( x^3 \)-terms: \( x^3 \)
- \( x^2 \)-terms: \( 3x^2 - 2x^2 \)
- \( x \)-terms: \( 9x + x \)
- Constant terms: \( 6 - 3 \)

Simplify each group:
- \( x^5 \)-terms: \( x^5 + 2x^5 = 3x^5 \)
- \( x^3 \)-terms: \( x^3 \)
- \( x^2 \)-terms: \( 3x^2 - 2x^2 = x^2 \)
- \( x \)-terms: \( 9x + x = 10x \)
- Constant terms: \( 6 - 3 = 3 \)

Combine all simplified terms:
\[
3x^5 + x^3 + x^2 + 10x + 3
\]

#### Final Answer for Part 2:
\[
\boxed{3x^5 + x^3 + x^2 + 10x + 3}
\]

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Part 3: Simplify \( (5x^2 + 3x^3 - x + 6) - (4x^2 - 8x^3 + 5x - 2) \)

#### Step 1: Distribute the negative sign
Distribute the negative sign across the second polynomial:
\[
(5x^2 + 3x^3 - x + 6) - (4x^2 - 8x^3 + 5x - 2) = 5x^2 + 3x^3 - x + 6 - 4x^2 + 8x^3 - 5x + 2
\]

#### Step 2: Combine like terms
Group the terms by their powers of \( x \):
- \( x^3 \)-terms: \( 3x^3 + 8x^3 \)
- \( x^2 \)-terms: \( 5x^2 - 4x^2 \)
- \( x \)-terms: \( -x - 5x \)
- Constant terms: \( 6 + 2 \)

Simplify each group:
- \( x^3 \)-terms: \( 3x^3 + 8x^3 = 11x^3 \)
- \( x^2 \)-terms: \( 5x^2 - 4x^2 = x^2 \)
- \( x \)-terms: \( -x - 5x = -6x \)
- Constant terms: \( 6 + 2 = 8 \)

Combine all simplified terms:
\[
11x^3 + x^2 - 6x + 8
\]

#### Final Answer for Part 3:
\[
\boxed{11x^3 + x^2 - 6x + 8}
\]

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Final Answers:

1. \( \boxed{6x^3 + 2x^2 - 5x - 10} \)
2. \( \boxed{3x^5 + x^3 + x^2 + 10x + 3} \)
3. \( \boxed{11x^3 + x^2 - 6x + 8} \)