Let's simplify the expression step by step:

Given expression:  
\[ x(x+2) - 3x(x+1) - 2(x^2 - x - 1) \]

### Step 1: Expand each term.

1. Expand \( x(x+2) \):
\[
x(x+2) = x^2 + 2x
\]

2. Expand \( -3x(x+1) \):
\[
-3x(x+1) = -3x^2 - 3x
\]

3. Expand \( -2(x^2 - x - 1) \):
\[
-2(x^2 - x - 1) = -2x^2 + 2x + 2
\]

### Step 2: Combine all the terms.
Now we substitute the expanded terms into the original expression:
\[
x^2 + 2x - 3x^2 - 3x - 2x^2 + 2x + 2
\]

### Step 3: Combine like terms.
- Combine \( x^2 \) terms:  
\[
x^2 - 3x^2 - 2x^2 = -4x^2
\]

- Combine \( x \) terms:  
\[
2x - 3x + 2x = x
\]

- The constant term is \( +2 \).

### Final expression:
\[
-4x^2 + x + 2
\]

This is the simplified expression.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How can you simplify expressions with polynomials of higher degrees?
2. What are some methods to factor quadratic expressions?
3. How can the distributive property help simplify complex expressions?
4. What are the common mistakes to avoid when expanding and combining like terms?
5. How do negative signs affect the distribution in algebraic expressions?

**Tip:** Always expand each term individually before combining like terms for accurate results.

Learn how to simplify the polynomial expression x(x+2) - 3x(x+1) - 2(x^2 - x - 1) with a detailed step-by-step solution. This problem covers key algebraic concepts, including the distributive property and combining like terms. Suitable for students in Grades 8-10.

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