Polynomial addition and subtraction problems for practice.

Let's solve each problem step by step. These are polynomial addition and subtraction problems. We will simplify each expression by combining like terms.

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1.) $(x^2 - 4x) - (x^2 - x)$

Distribute the negative sign:
$$
x^2 - 4x - x^2 + x
$$
Combine like terms:
$$
(x^2 - x^2) + (-4x + x) = 0 - 3x = -3x
$$
✔ Answer: $-3x$

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2.) $(2x^2 - 5x^3) + (5x^3 - 3x^2)$

Remove parentheses:
$$
2x^2 - 5x^3 + 5x^3 - 3x^2
$$
Combine like terms:
$$
(-5x^3 + 5x^3) + (2x^2 - 3x^2) = 0 - x^2 = -x^2
$$
✔ Answer: $-x^2$

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3.) $(2x^2 - 3x^3) - (x^3 - 5x^2)$

Distribute the negative sign:
$$
2x^2 - 3x^3 - x^3 + 5x^2
$$
Combine like terms:
$$
(-3x^3 - x^3) + (2x^2 + 5x^2) = -4x^3 + 7x^2
$$
✔ Answer: $-4x^3 + 7x^2$

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4.) $(n - 4n^2) + (3n^2 - n)$

Remove parentheses:
$$
n - 4n^2 + 3n^2 - n
$$
Combine like terms:
$$
(n - n) + (-4n^2 + 3n^2) = 0 - n^2 = -n^2
$$
✔ Answer: $-n^2$

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5.) $(4n^4 + 4) + (5 - n^4)$

Remove parentheses:
$$
4n^4 + 4 + 5 - n^4
$$
Combine like terms:
$$
(4n^4 - n^4) + (4 + 5) = 3n^4 + 9
$$
✔ Answer: $3n^4 + 9$

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6.) $(4n^3 + 2n^4) - (5n^4 - 4n^3)$

Distribute the negative sign:
$$
4n^3 + 2n^4 - 5n^4 + 4n^3
$$
Combine like terms:
$$
(2n^4 - 5n^4) + (4n^3 + 4n^3) = -3n^4 + 8n^3
$$
✔ Answer: $-3n^4 + 8n^3$

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7.) $(2n^4 - 5) + (1 - 3n^4)$

Remove parentheses:
$$
2n^4 - 5 + 1 - 3n^4
$$
Combine like terms:
$$
(2n^4 - 3n^4) + (-5 + 1) = -n^4 - 4
$$
✔ Answer: $-n^4 - 4$

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8.) $(4x^3 - 3x^2) + (3x^4 - 5x^2)$

Remove parentheses:
$$
4x^3 - 3x^2 + 3x^4 - 5x^2
$$
Rearrange and combine like terms:
$$
3x^4 + 4x^3 + (-3x^2 - 5x^2) = 3x^4 + 4x^3 - 8x^2
$$
✔ Answer: $3x^4 + 4x^3 - 8x^2$

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9.) $(3n - 2n^3) + (4n^3 + n)$

Remove parentheses:
$$
3n - 2n^3 + 4n^3 + n
$$
Combine like terms:
$$
(-2n^3 + 4n^3) + (3n + n) = 2n^3 + 4n
$$
✔ Answer: $2n^3 + 4n$

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10.) $(4x + 2) - (2 - x)$

Distribute the negative sign:
$$
4x + 2 - 2 + x
$$
Combine like terms:
$$
(4x + x) + (2 - 2) = 5x + 0 = 5x
$$
✔ Answer: $5x$

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✔ Final Answers:

1. $-3x$
2. $-x^2$
3. $-4x^3 + 7x^2$
4. $-n^2$
5. $3n^4 + 9$
6. $-3n^4 + 8n^3$
7. $-n^4 - 4$
8. $3x^4 + 4x^3 - 8x^2$
9. $2n^3 + 4n$
10. $5x$

Let me know if you'd like these explained in a different format or with visual steps!