Simplifying polynomial expressions by subtracting like terms.

Problem: Simplify the given expressions.

We will solve each expression step by step.

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#### Expression 11:
$$
(x^3 + 5 + 2x^2) - (3x - x^2 - 2x^4)
$$

1. Distribute the negative sign through the second set of parentheses:
$$
(x^3 + 5 + 2x^2) - (3x - x^2 - 2x^4) = x^3 + 5 + 2x^2 - 3x + x^2 + 2x^4
$$

2. Combine like terms:
- Terms with $x^4$: $2x^4$
- Terms with $x^3$: $x^3$
- Terms with $x^2$: $2x^2 + x^2 = 3x^2$
- Terms with $x$: $-3x$
- Constant terms: $5$

So, the expression becomes:
$$
2x^4 + x^3 + 3x^2 - 3x + 5
$$

Final Answer for Expression 11:
$$
\boxed{2x^4 + x^3 + 3x^2 - 3x + 5}
$$

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#### Expression 12:
$$
(a^2 + 4a + 8) - (a^2 + 2a + 2)
$$

1. Distribute the negative sign through the second set of parentheses:
$$
(a^2 + 4a + 8) - (a^2 + 2a + 2) = a^2 + 4a + 8 - a^2 - 2a - 2
$$

2. Combine like terms:
- Terms with $a^2$: $a^2 - a^2 = 0$
- Terms with $a$: $4a - 2a = 2a$
- Constant terms: $8 - 2 = 6$

So, the expression becomes:
$$
2a + 6
$$

Final Answer for Expression 12:
$$
\boxed{2a + 6}
$$

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#### Expression 13:
$$
(4b^3 + b^2 - 2) - (3b^3 + 6b + 2)
$$

1. Distribute the negative sign through the second set of parentheses:
$$
(4b^3 + b^2 - 2) - (3b^3 + 6b + 2) = 4b^3 + b^2 - 2 - 3b^3 - 6b - 2
$$

2. Combine like terms:
- Terms with $b^3$: $4b^3 - 3b^3 = b^3$
- Terms with $b^2$: $b^2$
- Terms with $b$: $-6b$
- Constant terms: $-2 - 2 = -4$

So, the expression becomes:
$$
b^3 + b^2 - 6b - 4
$$

Final Answer for Expression 13:
$$
\boxed{b^3 + b^2 - 6b - 4}
$$

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#### Expression 14:
$$
(6 - 2m + m^2) - (-4m^2 + 3m + 1)
$$

1. Distribute the negative sign through the second set of parentheses:
$$
(6 - 2m + m^2) - (-4m^2 + 3m + 1) = 6 - 2m + m^2 + 4m^2 - 3m - 1
$$

2. Combine like terms:
- Terms with $m^2$: $m^2 + 4m^2 = 5m^2$
- Terms with $m$: $-2m - 3m = -5m$
- Constant terms: $6 - 1 = 5$

So, the expression becomes:
$$
5m^2 - 5m + 5
$$

Final Answer for Expression 14:
$$
\boxed{5m^2 - 5m + 5}
$$

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Final Answers:

1. $\boxed{2x^4 + x^3 + 3x^2 - 3x + 5}$
2. $\boxed{2a + 6}$
3. $\boxed{b^3 + b^2 - 6b - 4}$
4. $\boxed{5m^2 - 5m + 5}$